She's not supposed to have ever been to Westeros. The story Dany is told of her childhood is that she was born on Dragonstone during a great storm, and that as a baby she was smuggled to Braavos, where she and Viserys lived until she was five. Her memory of a lemon tree apparently demonstrates that this story is wrong or incomplete.
As to what this means, we don't know, other than that there's something about Dany's past yet to be revealed. One possibility is that she is not really a Targaryen. Dany assumes throughout the series that she has the birthright to conquer Westeros, it's one of the cornerstones of her identity. Discovering that this is not true would require her to completely rebuild herself. Self identity is one of the major themes of the series, virtually every character struggles with it in some way (most obviously, Arya, Jon, Jaime, Tyrion, Brienne, Sansa but there are many others). A lot of people think that is super unlikely though, there are other possibilities and it could turn out to be something pretty minor.
There are a few other oddities around Dany's origin story, but they're much more tinfoilish than the clearly deliberate Lemongate. One thing is that when they're prepping her for Drogo, Illyrio sometimes talks like he might have his doubts that she's the genuine article, for instance:
Quote:
“She is a vision, Your Grace, a vision,” he told her brother. “Drogo will be enraptured.”
“She’s too skinny,” Viserys said. His hair, the same silver-blond as hers, had been pulled back tightly behind his head and fastened with a dragonbone brooch. It was a severe look that emphasized the hard, gaunt lines of his face. He rested his hand on the hilt of the sword that Illyrio had lent him, and said, “Are you sure that Khal Drogo likes his women this young?”
“She has had her blood. She is old enough for the khal, “ Illyrio told him, not for the first time. “Look at her. That silvergold hair, those purple eyes... she is the blood of old Valyria, no doubt, no doubt... and highborn, daughter of the old king, sister to the new, she cannot fail to entrance our Drogo.” When he released her hand, Daenerys found herself trembling.
This is slightly odd, because if she's a Targaryen, of course she looks Valyrian. Illyrio is like "dude don't worry, she looks obviously Valyrian, it's very convincing. Er... and I mean, she's the daughter of the king, of course". It's not at all conclusive of course but there are several spots like that where things feel a bit off. Elsewhere Illyrio talks about "years of planning" to marry Dany to Drogo, even though he's only supposed to have met Dany 6 months ago.
If you want to get into the weeds on Lemongate, the HAY GUYS LEMON TREES DON'T GROW IN BRAAVOS, MORESO IN DORNE quotes from the books are in an FAQ
here.