Why do some questions not start with an auxiliary verb?

When I learned English, my teachers told me that all questions must have an auxiliary verb at the beginning, just like Are you mad? or Is she playing? do.

But when watching some movies or talking with people who speak English, they just ask using things like You mad? and She’s playing?

Of course, the kind of intonation let me know they are asking a question, but why this happen?

They do start with an auxiliary verb, but since it's predictable, it's often omitted.

These are examples of what's called Conversational Deletion in the literature.
The link has references and further examples.

I can think of several valid examples of questions that neither start with an auxiliary verb, nor have been pruned through conversational deletion:

Come again? [Idiomatic question construction meaning "Please repeat whatever it was you just said", or sometimes merely expressing disbelief]

In what way? [Seeking some kind of clarification]

By when? [Seeking clarification of a deadline or other time constraint]

Really? [Seeking confirmation of a purported fact]

Meaning what? [Seeking clarification]

Because real English speakers speak real English, and didn't learn it from your teachers.

Forms like You hungry? and She coming? are common in speech, but are very informal, and would not be found in most written contexts.