The most important part of that conversation would be:
Brienne: I can go and find the master of arms for you, my lady.
Arya: He didn't beat the Hound. You did.
My own interpretation was that given Arya's character, it can be read in a combination of (at least) three ways:
Arya wants to improve
Throughout the books/series she's been continually pushing herself. It started by badgering Ned until he'd let her train with Syrio Forel. Even after leaving King's Landing, and generally going through hell, she kept training by herself.
After she met the Faceless Man, she went through an awful lot of trouble to learn their skills.
In the context of her jumping at every chance to make herself stronger, training with the fighter who beat the Hound (a brutal and effective killer) makes perfect sense.
Brienne killed the Hound
In case anyone did not notice yet: Arya holds grudges. The Hound is a bit of a special case though: while he holds a place on her (s)hit list for killing the butcher's boy, he also was her companion for quite some time.
For both reasons, she'd be pissed someone else killed him. Her attitude to conflicting feelings generally seems to be to tackle them head on.
It's also not unlikely that in her mind, fighters are measured by how they'd stack up against the Hound. Beating Brienne indirectly means she's beating the Hound.
Arya wants to establish herself
In my opinion the weakest of the interpretations, but worth a mention. She's always been the little, useless sister to Sansa. Kicking ass and taking names is one way to point out that little sister ceased to be useless. Bonus points for giving the middle finger to all those ladylike skills she's always despised.