The basic purpose of a latch is to save the state of some particular input. The latches from today's lecture are too simple to do much by themselves, but they can be used as part of more complicated circuits.
As an example from my own research: I might want to know which electronics channels fired when a particle went through my detector. I could design a circuit to latch the signals on each channel when a trigger condition was met, so that I could later read out the pattern of channels that fired.
There are lots of other applications for which one might want to save digital information. In general, latches and flip-flops are used to save the states of the bits that encode binary numbers (one wants one's computer to remember the numbers it computes or it isn't much use...)