Teeming means completely full, especially with living things. If your grandmother's apartment is teeming with cats, she sure has a lot of them.
Any time something (or someplace) is filled with life, it's teeming with it. A gorgeous purple garden in the south of France might be teeming with lavender plants, and you might describe the crowded state fair grounds as teeming with people. The Old English root teman means to give birth to, and this used to be a common meaning of teeming — being fertile or pregnant, or producing young.