Two objects are dropped from a bridge an interval of 1.0s apart. Air resistance is negligible. During the time that both objects continue to fall, their separation increases.

Can someone explain why it increases, instead of staying constant?

Their separation would be constant if they were both falling at the same speed. But they're not. A falling object falls 9.8 m/s faster every second than it fell in the previous second.

So the one that dropped first ... and had an extra second to fall ... is always falling faster than the one that dropped later. So it keeps getting farther ahead.