The Kauffman bracket skein algebra is one of the central objects of study in quantum topology. It is a noncommutative algebra that quantizes the SL(2,C)-character variety of a surface. In this talk, I will describe a new way of thinking about this algebra. The key insight is that skein algebras are closely related to objects called quantized K-theoretic Coulomb branches, which originate in mathematical physics and geometric representation theory. I will explain how, when the underlying surface has genus zero, this leads to a monoidal categorification of the skein algebra, partially answering a question of D. Thurston from 2014. This talk is based on joint work with P. Shan and H.K. Kim.