Rare earths are mined like most other metals. Hard-rock deposits such as bastnaesite and monazite are drilled, blasted, and hauled out of an open pit. The ore is not remarkable to look at; the rare earth minerals are a small fraction of the rock, and the rest is waste that will be removed in the next step.
Ionic clay deposits are the exception. The rare earths are held loosely on the surface of fine clay particles, so there is no rock to crush. The clay is already in a form ready for extraction, and its element mix tends to be more favorable for magnet making, with a larger share of the heavy rare earths that magnets need.