Actually it's well established in the Books how and why Arya ended in The House of Black and White, Braavos.
Right after the Hound gets a mortal wound and Arya parts with him, she arrives in a small town Saltpans. She wants to make it to the Wall, because Jon Snow's there. She buys her passage on a Braavosi galleas with the iron coin given to her by Jaqen. But the captain refuses to sail to the North, so instead she ends up in Braavos. In Braavos she visits the house of the Black and White and seeks Jaqen to learn the art of being assassin. And she wants blood and revenge.
Now the long answer, with the quotes from the book.
ASOS, chapter 74, Arya.
“I want to see the captain,” Arya told him. He only shouted louder.
But the commotion drew the attention of a stout grey-haired man in a
coat of purple wool, and he spoke the Common Tongue. “I am captain
here,” he said. “What is your wish? Be quick, child, we have a tide to
“I want to go north, to the Wall. Here, I can pay.” She gave him the
purse. “The Night’s Watch has a castle on the sea.”
“Eastwatch.” The captain spilled out the silver onto his palm and
frowned. “Is this all you have?”
It is not enough, Arya knew without being told. She could see it on
“Wait,” Arya said suddenly. “I have something else.” She had stuffed
it down inside her smallclothes to keep it safe, so she had to dig
deep to find it, while the oarsmen laughed and the captain lingered
with obvious impatience. “One more silver will make no difference,
child,” he finally said.
“It’s not silver.” Her fingers closed on it. “It’s iron. Here.” She
pressed it into his hand, the small black iron coin that Jaqen H’ghar
had given her, so worn the man whose head it bore had no features.
It’s probably worthless, but . . .
The captain turned it over and blinked at it, then looked at her
again. “This . . . how . . . ?”
Jaqen said to say the words too. Arya crossed her arms against her
chest. “Valar morghulis,” she said, as loud as if she’d known what it
“Valar dohaeris,” he replied, touching his brow with two fingers. “Of
course you shall have a cabin.”
That's how Arya bought her passage. But to what place, that's revealed in the next chapter.
AFFC, chapter 6, Arya
Arya stood at the prow, one hand resting on the gilded figurehead, a
maiden with a bowl of fruit. For half a heartbeat she let herself
pretend that it was her home ahead.
But that was stupid. Her home was gone, her parents dead, and all her
brothers slain but Jon Snow on the Wall. That was where she had wanted
to go. She told the captain as much, but even the iron coin did not
sway him. Arya never seemed to find the places she set out to reach.
Yoren had sworn to deliver her to Winterfell, only she had ended up in
Harrenhal and Yoren in his grave. When she escaped Harrenhal for
Riverrun, Lem and Anguy and Tom o’ Sevens took her captive and dragged
her to the hollow hill instead. Then the Hound had stolen her and
dragged her to the Twins. Arya had left him dying by the river and
gone ahead to Saltpans, hoping to take passage for
Eastwatch-by-the-Sea, only . . .
Braavos might not be so bad. Syrio was from Braavos, and Jaqen might
be there as well. It was Jaqen who had given her the iron coin. He
hadn’t truly been her friend, the way that Syrio had, but what good
had friends ever done her?
That's how she ends up in Braavos.
And again in the same chapter when she enters the House of Black and White, she has the following conversation with a old man, later called "the kindly man".
“What place is this?” she asked him.
“A place of peace.” His voice was gentle. “You are safe here. This is
the House of Black and White, my child. Though you are young to seek
the favor of the Many-Faced God.”
“Is he like the southron god, the one with seven faces?”
“Seven? No. He has faces beyond count, little one, as many faces as
there are stars in the sky. In Braavos, men worship as they will . . .
but at the end of every road stands Him of Many Faces, waiting. He
will be there for you one day, do not fear. You need not rush to his
“I only came to find Jaqen H’ghar.”
“I do not know this name.”
Her heart sank. “He was from Lorath. His hair was white on one side
and red on the other. He said he’d teach me secrets, and gave me
this.” The iron coin was clutched in her fist. When she opened her
fingers, it clung to her sweaty palm.
She's there for Jaqen, and she wants to learn. And as for her motivation, again in this very chapter.
“Do you fear death?”
She bit her lip. “No.”
“Let us see.” The priest lowered his cowl. Beneath he had no face;
only a yellowed skull with a few scraps of skin still clinging to the
cheeks, and a white worm wriggling from one empty eye socket. “Kiss
me, child,” he croaked, in a voice as dry and husky as a death rattle.
Does he think to scare me? Arya kissed him where his nose should be
and plucked the grave worm from his eye to eat it, but it melted like
a shadow in her hand.
The yellow skull was melting too, and the kindliest old man that she
had ever seen was smiling down at her. “No one has ever tried to eat
my worm before,” he said. “Are you hungry, child?”
Yes, she thought, but not for food.
She wants revenge. She's hungry for the blood.