That's an interesting direction; I've always been told to play closer to the outside edge of the keys! But then again, I'm a skinny dude with skinny fingers, so I guess we have opposite problems.
I'd suggest taking the middle ground; make sure your fingers are curved, and keep the fingertips just outside the edge of the black keys. You should be able to hit both white and black keys without any significant wrist motion forwards or backwards, by stretching your finger out. I suspect that this was the instruction that your teacher was trying to get at, but I can't be sure.
It's really important to curve your fingers (it shouldn't be as curved as while holding a can of
beer soda; rather, it should be a natural curve, like what happens when you relax your hand after making a fist). If you play flat-fingered, you'll need to move your wrists a little for each note you play, which will makes playing much harder. It's also important because letting your wrists flow smoothly helps with playing a consistent legato, even spanning white and black keys. The curve helps prevent strain in your wrist, too. Too much strain could lead to RSIs like carpal tunnel syndrome. But i'm starting to babble again. Your fingers should curve very naturally if you keep your wrists above the level of the keys and let your fingers relax.
To summarize: Fingers should hit the keys somewhere in the broad area of the white keys, and as near to the forward edge of the black keys as possible. If your fingers hit somewhere in the skinny part of the white keys, they're probably too far forward (I say probably because there are a handful of chords that can't be formed any other way). Fingers should be curved naturally. Wrists should move smoothly (although, now that I think about it, that's entirely irrelevant for this question).