When you liaise with someone, you meet up or connect with them, usually so you can work together on something cooperatively. Two rival soccer teams might liaise to discuss the muddy conditions of the town's soccer fields.
Liaise came from the word liaison during World War I, when it was used mostly to mean "form a military cooperation or alliance." The Latin root is ligatio, or "a binding"; when two groups liaise, they bind together or connect in order to gain a result that they both want. The very trickiest thing about the word liaise is remembering that it's spelled not with one "i" but with two.