to sprout up out of the ground, or to make something sprout: the fresh springing of new growth
This verb pictures new life pushing up out of the earth. In its plain form it is what a seed, a plant, or a field does on its own: the shoot breaks the surface and comes into view. It also has a paired sense where someone or something makes that growth happen, most often God sending the plants up or the watered soil bringing them forth. The two are the same event seen from opposite sides, the sprouting and the causing of it, which is why one Hebrew word covers both.
From that farming picture the word stretches naturally to anything that grows the same way. Hair and a beard grow back after they are cut. Trouble is said to spring from the ground like a weed. And in the prophets the word turns into a favorite image for God bringing new things into being: rescue, right living, and honor are described as springing up from the earth the way a crop does, sudden, visible, and unstoppable once God starts it.
The most weighted use is when a king or deliverer is said to spring up. God promises to make a horn (a symbol of strength) sprout for David, and a coming ruler is announced with this same verb. That prophetic thread is where the related noun, the messianic title usually translated Branch or Shoot, comes from, though the noun is a separate word from this verb.