What is this insect flying in my Los Angeles bathroom?
Looks like the common clothes moth (Tineola bisselliella). And, yes, the larvae (caterpillars) are likely the source of the holes in your shirts. You can purchase traps online or from hardware stores, or you can go the pesticide route if necessary (e.g., Delta Dust is available from online retailers).
See this site for further info, or search the web for more info on this species. https://en.wikipedia.org/wiki/Tineola_bisselliella