Take a look at your typical sports shoes. Chances are they aren't flat i.e. they have heels. When you're wearing shoes with heels, your body tilts slightly forward throwing you a bit off balance especially when you're squatting. Especially when you're front squatting.
This was made more apparent recently when I'm training at a public RM2 per entry gym over in Bangsar. I usually take off my shoes when I'm doing my lifts but one fine day, this obnoxious little man who frequents the gym said that taking off our shoes are prohibited.
The humble Chuck Taylor/Converse shoes
Not wanting any trouble, I squatted with my shoes on and I feel extremely uncomfortable performing them. I felt like I could tilt forward at any time. Apparently quite a number of trainers know about this and choose to either train with no shoes on or wear flats like the Chuck Taylors or Vibram Five Fingers. Me, I just like to train barefoot. Much cheaper.
The funkier Vibram Five Fingers
The ever reliable and cheap.....
Hmmm... and all this while I thought martial artists train with no shoes on because they don't want to hurt their sparring partners. Turns out it was because it's more stable that way. Interesting.