How to show maturity specifically to med school admissions?
Have a record of good grades -- shows academic ability and consistency in managing academic work, over time. A strong upward trend in grades demonstrates maturity, had you made mistakes in the past and learned from them.
Have some leadership experience and be able to talk about what difference you made in those positions and what skills you learned or applied.
Show in your essays and interview that you can speak cogently about your goals, interests, motivations, strength and weaknesses. Basically, a mature applicant knows themselves and this comes through to the admissions people. Learn to communicate effectively and respectfully about the needs of others. Volunteering is good for this.
When you draft essays, as well as having people read for content and grammar, also have them rate how mature it reads, because you can come off as more or less mature in your writing style.
That's what I could think of. Good luck!