How does an annular tear typically occur?

Twisting. Movements, especially twisting cause microscopic tears in the collagen fibers (tiny structural elements like the fibers or threads of a fabric) of the annulus (part of the disk). The body has some ability to heal such tears, but there is a limit. When the amount of microscopic tears becomes large, the tears connect, causing a larger tear. Sometimes those tears are painful.
Due to aging. Most tears to the outer layer of the disc are due to aging or degenerative changes & not painful. They can be due to injury or overload of the disc without the central disc material (nucleus pulposus) leaking out (herniation) & they can be quite painful as the only nerves in the disc are in this outer layer. Most heal but the tear may still be apparent on a MRI long after the pain has resolved.
Aging and trauma. Annular tears generally occur secondary to the aging process. As the disc loses its water content tears and failures in the annulus can occur. Trauma is another cause of annular tears.