Good, Bean!
Anyway, to exactly define what a compact space is, it is necessary a small course in Topology, so to have the correct definitions of topological space, open and closed set, and so on... and also, the usage of these words is NOT the usual one, for example, OPEN does not mean NOT CLOSED.
So, I don't hope that it is really possible to explain the meaning of TO COMPACTIFY to everyone, unless giving some confused - and maybe even partially wrong - idea..

Emanuela