If you are running two mixers at once - perhaps one for the monitor system and one for main PA, or one for broadcast and one for PA, or a performer wants his/her mic split into a powered monitor and the PA - you need the ability to split the signal from the source so that it can go both ways. It is important that the output of the source encounters the right impedance. For example if you simply used a "Y" adaptor, signal strength and quality would be lost. For this reason, a professional splitter will have a built-in transformer. Since you may be splitting the balanced output of an instrument amp or preamp, another consideration is "hum" caused by a ground loop.
A good splitter box will also feature a ground lift switch to reduce that problem.