Aside from “warming up” the brain (rediscovering that deeply energetic coordination that enables good singing), the vocal warm-up is about activating phonatory, articulatory and support-related muscles.
Know that as muscle fibers get warm, they become more fluid, so that they stretch and contract more rapidly. Both florid and sustained singing (florid singing on fewer and longer notes…more on that topic later) become easier. The rapidity factor is important, since the vocal cycle happens on average 100-500 times per second for men and approximately 300-900 times for women. Related muscles must be very responsive, indeed, for such a trick!
The core, supporting muscles of the torso and lower body must also be able to vary the intensity of their engagement nimbly and sympathetically, in order for the vocal apparatus to work at peak efficiency and freedom. For this reason, simple breath-related exercises (even non-vocal) can be helpful in warming up the body. What we call “support” or “appoggio” in Classical singing has to do not only with breath management, but with providing stability for the body, thus allowing easier engagement of the articulators, and enhancing the ease of performance on every level. In addition to these low support muscles, the vocal folds themselves and related muscles of the upper body need some warm-up time to reach optimum function.
Athletes in various sports require warm-ups of varying character, intensity and length; some athletes seem to reach peak (or at least functional) level more quickly than others. I think of the pinch-runner in baseball, who jogs out to first base on very short notice and does a few stretches on the spot. To be sure, that runner has executed a more complete and generous warm-up a few minutes or hours earlier.
Some singers warm up very quickly — so much so, that it may seem no warm-up exercises are required. Depending on the repertoire, it may be true that a well-functioning speaking voice will sufficiently enable the desired result. I have heard singers say that they don’t need to warm up; in my experience, Classical singers who make such a claim are usually basses. Other singers more nearly “grind down” their voice, instead of warming it up, by vocalizing too much, too aggressively, and/or with inappropriate thought.
Though some of us can reach a functional level of singing almost immediately (often depending on time of day and energy level), a singer typically finds that after a few minutes of singing, he/she will reach a higher level of comfort, flexibility and power. Each of us must determine how much warm-up is enough, and must never expect technical mastery, intelligence or extra effort to compensate for an ineffective warm-up.