FE:A has a compelling story, with so many well-written characters. Each of them has a unique, distinct personality that you can't help but love (even the weird ones). Being something of a hopeless romantic, my favourite part of the game has to be the support options, which play a bigger role than you might think.
Given that the children who come from the future bear a distinct resemblance (Hair colour, occasionally class) to whomever you marry their mother (or father, in Female!Morgan's case) to, not to mention stats, it's something to pay attention to, even if you don't like the quirky character dialogue.
That said, the battles themselves are also quite challenging. Between the weapon triangle, winged units' vulnerability to archers, weapon lifespan, and reinforcements that never stop coming, it's a task all its own to keep your units in one piece, let alone complete the battle objective.
Supports and story aside, I also enjoy the wireless functions - spot-pass battles, fighting against teams from previous games, I live for it. It's a nice bit of nostalgia to fight alongside characters like Marth, Roy, Hector, etc...
Visually, the game looks better than anything I would have expected from a hand-held console (even if it is kind of hilarious that the characters don't have feet outside of the cinematic cutscenes), and the cutscenes are glorious in 3D, and 10/10 voice acting from all involved.
All in all, definitely one of my favorite games of all time. I would definitely recommend it if you're into strategy games, or even just for the story itself.