This was done because the cafeterias were spending a lot of money washing trays. No trays, no cost to wash them. Instant cost savings!
At the link above, you can hear the full article and the opinions of various affected parties (both pro and con).
In the light of the removal of remote dish pickup at my workplace, and given that one of the food service companies mentioned in the NPR article runs the cafeteria at work, I'm sure we'll see trays going away fairly soon.
There was a side effect that took place at the colleges: Students took less food, ate less food, and there was less food waste. Hmmmm...