ood breathing and breath control when playing the saxophone is important for two reasons:

Playing extended phrases without running out of breath
Having good breath support from your lungs

The second of these is probably the most important as it can actually help your tone. It is a well established technique to keep your throat open while playing (an open throat is what happens when you yawn, as opposed to a closed throat which happens when you cough). This is important for singers as well as wind players as a large part of what shapes the sound is the physical dimensions of the inside of your mouth, throat and possibly lungs and nasal cavities. But being able to control the pressure of air in your lungs, while keeping your throat open, is not something people do naturally. Normally the air just rushes out unless you close your throat to stop that happening. The trick is to use a muscle underneath your lungs to control that natural tendency for the lungs to contract and expel the air all at once. This is the diaphragm.