How do academics with teaching responsibilities, etc. find the time to do research?
How does anyone "find the time" to do anything? The answer is that mathematicians don't "find the time", the time is already there and they make use of it. Your premise that those other activities you mentioned already fill up 100% of the time is simply incorrect. In particular, summers, breaks and other periods with no teaching offer a convenient and relatively distraction-free environment for doing research. But even teaching is not mutually exclusive with doing some research.