Learn Russian: Он сорвал зелёное яблоко с дерева.

In Russian, prepositions of direction work in pairs. The preposition с (from/off of) is the exact opposite of на (on/onto). Since an apple grows on a tree (на дереве), taking it from or off the tree requires с (с дерева). If the apple were inside a box (в коробке), you would take it out using из (из коробки).

Is it a coincidence that the verb сорвал starts with со- and the preposition is с?

Not at all! The verbal prefix с- or со- often indicates movement away from a surface, which pairs perfectly with the preposition с. The base verb is рвать (to tear), so сорвать literally means to tear off, and he tore it off the tree.

яблоко is the direct object being picked. Why hasn't its ending changed?

яблоко is a neuter noun, and for all neuter inanimate nouns, the Accusative case (used for direct objects) is identical to the dictionary (Nominative) form. The same rule applies to the adjective modifying it, leaving it as зелёное.

What is the dictionary form of дерева, and why does it end in -а here?