Aguantar means to bear. The Diccionario del español de México gives, among its senses, to tolerate with patience someone who exasperates, as in aguantar al jefe, to put up with the boss. Aguantar vara or aguantar mecha means “soportar con tranquilidad, buen ánimo o entereza situaciones difíciles, adversas, molestas” (to bear difficult, adverse or annoying situations calmly, in good spirits or with fortitude).
In 2019, the linguist Carmen Teresa Fajardo Rojas of El Colegio de México studied the variants aguantar bala, aguantar mecha and aguantar vara in the journal Lingüística Mexicana. She found that aguantar vara is the most vital of the variants that carry this meaning.