A sprout is a small growth on a plant — a little new bud. Other things can sprout too: kids are constantly sprouting (growing).
The key thing to think of when you're trying to remember the meaning of sprout is growth — as a noun, a sprout is a new growth of a plant, and as a verb, to sprout means to grow. Sprouting mainly applies to height and to the young, whether you're talking about plants, people, or things. An older person who gains fifty pounds is growing but not sprouting.