Angular momentum is a fundamental quantity of Nature, and its conservation is used to understand the formation and shape of galaxies (notice the farther out a star is the slower it moves), how the planets orbit the sun, hurricanes and tornadoes, spinning gymnasts and divers and figure skaters, rotating and collapsing stars, and you sitting on rotating chairs. Oh, keep in mind that 2 of the 4 quantum numbers you learn in chemistry are angular momenta. This is an important quantity!
Here are a couple examples of how to apply conservation of angular momentum. One version is like a kid crawling on a rotating playground disk, where we use the version L = Iw. The moment of inertia changes, so the angular velocity must change. The second version is like a pinball machine, where a flap (rotating stick) makes a collision with a ball (point mass), and we need to figure out the speeds after the collision. See if the application of the conservation law makes sense for these and similar problems.