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Since the Doom of Valyria, it has been shunned by nearly all of humankind.

A few people have, however, tried to travel to Valyria, none lived to tell the tale. We have the self-proclaimed Emperor Aurion who went to reclaim Valyria with an army, he was never heard from again even though he had a dragon.

Then we have Volantis who sent a fleet to reclaim Valyria too, the fleet was lost with all hands. Then we have King Tommen II Lannister who disappeared along with the Lannister ancestral sword Brightroar in his quest to plunder the ruins of Valyria for knowledge, he met the same fate. Last of all, we have Tywin Lannister's brother Gerion who set out on a quest to find the lost Brightroar, he too went missing.

With each campaign meeting the same end, we have the curious case of Euron Greyjoy who claims that not only did he sail to Valyria but also brought back relics such as a dragon-binding horn and Valyrian Steel armour to prove it. He even claims to have found a dragon-egg. So how did he succeed where others before him failed?

Canon Answer

There's nothing explicit in canon as of now to confirm or reject Euron's claims. We just have Euron's word for it and the relics as proof of his claims. Some of the characters in universe are not convinced of his claims, while most of his supporters appear to be.

Speculation

It is very likely that Euron is lying that he went to Valyria. Let's look back at the previous attempts.

Aurion went with a huge army and a dragon and just disappeared. Even his dragon couldn't protect him from whatever curse is hanging over the ruins. Volantis sent an expedition at the zenith of their power and the whole fleet was lost. The same fate befell the Lannister King. One can't even assume that Euron's small party, presumably just Silence and her crew with a few captured vessels, escaped the notice of whatever horrors dwell there because Gerion Lannister's small expedition was never seen again either.

Euron, however, has some curious relics with him such as the Dragonhorn and the Valyrian steel armour. No one else in the entire world has such an armour or such a horn as far as we know. But that doesn't mean that he must have found them in Valyria. He could have found them aboard some ship he looted or anywhere in Essos really given that Valyria spanned over most of Western Essos and there are Valyrian relics all over the free cities.

While Euron likes to boast about his expedition a lot, he quickly becomes angry and evasive if someone questions it.

“When?” The voice was Lord Rodrik’s. “When shall we return, Your
Grace? A year? Three years? Five? Your dragons are a world away, and
autumn is upon us.” The Reader walked forward, sounding all the
hazards. “Galleys guard the Redwyne Straits. The Dornish coast is dry
and bleak, four hundred leagues of whirlpools, cliffs, and hidden
shoals with hardly a safe landing anywhere. Beyond wait the
Stepstones, with their storms and their nests of Lysene and Myrish
pirates. If a thousand ships set sail, three hundred may reach the far
side of the narrow sea... and then what? Lys will not welcome us, nor
will Volantis. Where will you find fresh water, food? The first storm
will scatter us across half the earth.

A smile played across Euron’s blue lips. “I am the storm, my lord. The
first storm, and the last. I have taken the Silence on longer voyages
than this, and ones far more hazardous. Have you forgotten? I have
sailed the Smoking Sea and seen Valyria.”

Every man there knew that the Doom still ruled Valyria. The very sea
there boiled and smoked, and the land was overrun with demons. It was
said that any sailor who so much as glimpsed the fiery mountains of
Valyria rising above the waves would soon die a dreadful death, yet
the Crow’s Eye had been there, and returned.

“Have you?” the Reader asked, so softly.

Euron’s blue smile vanished. “Reader,” he said into the quiet, “you
would do well to keep your nose in your books.”

Victarion could feel the unease in the hall. He pushed himself to his
feet. “Brother,” he boomed. “You have not answered Harlaw’s
questions.”

Euron shrugged. “The price of slaves is rising. We will sell our
slaves in Lys and Volantis. That, and the plunder we have taken here,
will give us sufficient gold to buy provisions.
_{AFFC - The Reaver}

Note how his reaction and even when urged by his brother publicly, he merely answered the logistical questions and did not talk about his Valyrian adventure.

The recently released Fire & Blood provides us more information about someone who almost certainly visited Valyria and came back. What befell that person should more than rubbish any claims Euron might have.

Princess Aerea was the daughter of Queen-Dowager Rhaena and for a time heir to her uncle King Jaehaerys I. She quarrelled with her mother, took Balerion the Black Dread (largest and oldest dragon in the world at that time) for her mount and disappeared. Her mother and her uncle searched for her everywhere but could not find any trace of the girl. Eventually the King just gave up the search while Queen Rhaena kept looking for her everywhere in Westeros.

After a few years, the Black Dread reappeared over King's Landing and on his back was a skinny starved girl whose clothes were in tatters. When the dragon landed in the Red Keep, the Princess only managed to say "Please, I never" and collapsed. Ser Lucamore Strong, a Knight of the Kingsguard who took the Princess to her uncle testified that the girl was burning with a fever so high that he could feel it in his armour. He also claimed that there was something moving within her (King Jaehaerys later forbid him from mentioning it ever again). The most stunning thing was that the Black Dread himself had wounds and gashes all over him. What horrible thing could do that to the world's largest dragon? Anyhow, The Grand Maester and his staff tried their best to save her but she perished within the day.

Septon Barth provides us a lot of details into what happened to her. She was literally burning and Barth claims to touch her skin was as if to put your hand into a pot of boiling oil. Her lips and eyes were bleeding and there were creatures moving within her. Her skin grew darker and darker by the hour and started cracking as if she was burning from inside until it looked like pork cracklings. There was smoke coming out of all her cavities and then her eyes burst. The Septon and Grand Maester put her in an ice cold bath which stopped her heart due to shock of the immersion. And then serpent like creatures came out of her, serpents with hands, faces and mouths. They died immediately on contact with ice.

And he believed that while they had all been wondering where had Aerea taken the Black Dread, they should have instead asked themselves where did Balerion take Aerea? The dragon was the only living thing who had seen Valyria alive, when he came with Aenar the Exile came to Dragonstone. When the inexperienced Aerea took the dragon for her ride, she failed to control him and the dragon took her to his home; Valyria. That's the only destination which explains the fact that no one in Westeros or Essos saw the Black Dread although there were reports that a Black Dragon was fighting in pits of Meereen but Septon Barth dismissed them as rumours.

“Though I can offer no certain proof, I can suggest an answer. It
seems to me that we have all been asking the wrong question. Aerea
Targaryen was still well shy of her thirteenth nameday on the morning
she slipped from her mother’s castle. Though no stranger to dragons,
she had never ridden one before…and for reasons we may never
understand, she chose Balerion as her mount, instead of any one of the
younger and more tractable dragons she might have claimed.Driven as
she was by her conflicts with her mother, mayhaps she simply wanted a
beast larger and more fearsome than Queen Rhaena’s Dreamfyre.

It might also have been a desire to tame the beast that had slain her father
and his own dragon (though Princess Aerea had never known her father,
and it is hard to know what feelings she might have had about him and
his death). Regardless of her reasons, the choice was made. “The
princess might well have intended to fly to King’s Landing, just as
her mother suspected. It might have been her thought to seek out her
twin sister in Oldtown, or to go seeking after Lady Elissa Farman, who
had once promised to take her adventuring. Whatever her plans, they
did not matter. It is one thing to leap upon a dragon and quite
another to bend him to your will, particularly a beast as old and
fierce as the Black Dread.

From the very start we have asked, Where
did Aerea take Balerion? We should have been asking, Where did
Balerion take Aerea? “Only one answer makes sense. Recall, if you
will, that Balerion was the largest and oldest of the three dragons
that King Aegon and his sisters rode to conquest. Vhagar and Meraxes
had hatched on Dragonstone. Balerion alone had come to the island with
Aenar the Exile and Daenys the Dreamer, the youngest of the five
dragons they brought with them. The older dragons had died during the
intervening years, but Balerion lived on, growing ever larger,
fiercer, and more willful. If we discount the tales of certain
sorcerers and mountebanks (as we should), he is mayhaps the only
living creature in the world that knew Valyria before the Doom. “And
that is where he took the poor doomed child clinging to his back. If
she went willingly I would be most surprised, but she had neither the
knowledge nor the force of will to turn him.

“What befell her on
Valyria I cannot surmise. Judging from the condition in which she
returned to us, I do not even care to contemplate it. The Valyrians
were more than dragonlords. They practiced blood magic and other dark
arts as well, delving deep into the earth for secrets best left buried
and twisting the flesh of beasts and men to fashion monstrous and
unnatural chimeras. For these sins the gods in their wroth struck them
down. Valyria is accursed, all men agree, and even the boldest sailor
steers well clear of its smoking bones…but we would be mistaken to
believe that nothing lives there now. The things we found inside Aerea
Targaryen live there now, I would submit…along with such other horrors
as we cannot even begin to imagine. I have written here at length of
how theprincess died, but there is something else, something even more
frightening, that requires mention: “Balerion had wounds as well. That
enormous beast, the Black Dread, the most fearsome dragon ever to soar
through the skies of Westeros, returned to King’s Landing with
half-healed scars that no man recalled ever having seen before, and a
jagged rent down his left side almost nine feet long, a gaping red
wound from which his blood still dripped, hot and smoking.

“The lords
of Westeros are proud men, and the septons of the Faith and the
maesters of the Citadel in their own ways are prouder still, but there
is much and more of the nature of the world that we do not understand,
and may never understand. Mayhaps that is a mercy. The Father made men
curious, some say to test our faith. It is my own abiding sin that
whenever I come upon a door I must needs see what lies upon the
farther side, but certain doors are best left unopened. Aerea
Targaryen went through such a door.”
_{Fire and Blood - The Long Reign}

If there are creatures so mighty in Valyria who can fight, wound and force to fly even the Black Dread, and horrors beyond our imagination in Valyria, we can safely call Euron a liar. There's no way he could have succeeded with his longboat where Emperor Aurion and the Black Dread failed.