A proximity fuse is a fuse that is designed to detonate an explosive automatically when close enough to the target to destroy it. By sending out radio waves that are reflected by the target and comparing the frequency of the outgoing waves to the incoming waves, a radio proximity fuse uses the Doppler effect to determine its proximity to a target. When the incoming waves sharply decrease in frequency, the target is at its closest possible position.
Before their invention, detonation had to be induced either by direct contact, time since launch, or height. All of these have significant disadvantages. Getting direct contact with a relatively small moving target is hard (even ignoring the effect of wind); to set a time- or height-triggered fuse one must measure the height of the target (or even predict the height of the target at the time one will be able to get a missile in its neighbourhood).
With a proximity fuse, all one has to worry about is getting a shell or missile on a trajectory that, at some time, will pass close by the target (that still is a significant problem).