"To do something under the wire" means "to do something at the last possible moment".
The phrase "slipped it in" can have various meanings. In this case, it probably refers to submitting a piece of work / essay very close to the deadline.
I don't know where the phrase "under the wire came" from. Maybe someone else can explain?
As for the phrase "slipped it in": People are sometimes described as "slipping in" through doorways and windows when there is only a small gap, or the door/window is in the process of being closed. In the case of a submitting a piece of work, the meaning is very similar: there is only a short amount of time remaining, the "window of opportunity" ( <- a very common metaphor) is very small.