First off, Luke is a dreamer. He wants adventure. He wants to get out and experience things.
From The Empire Strikes Back
Yoda: This one a long time have I watched. All his life has he looked away, to the future, to the horizon. Never his mind on where he was, hmm? What he was doing. Hmm. Adventure. Heh. Excitement. Heh. A Jedi craves not these things.
Then there's the fact that he's just plain lonely and bored.
From A New Hope:
Luke chuckled sardonically and assumed a mocking pose. “If there’s a bright center to this universe, you’re on the world farthest from it.”
Luke later describes Tatooine again.
From the novelization and original script of A New Hope (cut from the final versions of the movie)
Believe me, Tatooine is a big hunk of nothing.”
The conversation continues a bit later. His best friend Biggs has already left and joined the Academy. He's back visiting.
Luke looked away, embarrassed. “Things haven’t exactly been the same since you left, either, Biggs. It’s been so—” Luke hunted for the right word and finally finished helplessly, “— so quiet.” His gaze traveled across the sandy, deserted streets of Anchorhead. “It’s always been quiet, really.”
And it's not like Tatooine is exactly safe these days.
Again, from the novelization:
Luke snorted derisively. “Not likely. I had to withdraw my application.” He looked away, unable to meet his friend’s disbelieving stare. “I had to. There’s been a lot of unrest among the sandpeople since you left, Biggs. They’ve even raided the outskirts of Anchorhead.”
At this point, Luke feels trapped, maybe a bit abandoned. He wants to get out to do things like his friend Biggs.
Silence hung in the air of the dining room after Luke departed. Husband and wife ate mechanically. Eventually Aunt Beru stopped shoving her food around her plate, looked up, and pointed out earnestly, “Owen, you can’t keep him here forever. Most of his friends are gone, the people he grew up with. The Academy means so much to him.”