Learn Russian: Несмотря на то, что вредителей много, этот сорт даёт хороший урожай. - Despite the fact that there are many pests, this variety yields a good harvest.

Несмотря на то, что is a compound conjunction meaning "despite the fact that". In Russian, you must use a comma to separate the clauses. The comma usually goes right before что to introduce the subordinate clause.

Can I just use несмотря на instead of the whole phrase несмотря на то, что?

Not in this sentence. Несмотря на (despite) must be followed directly by a noun or pronoun (e.g., "despite the pests"). Because we are introducing a full clause with its own subject and predicate ("there are many pests"), we must use the full connector несмотря на то, что (despite the fact that).

Why is вредителей in the genitive case?

It is in the genitive plural because it is being quantified by the word много (many / a lot of). The dictionary form is вредитель (pest).

Is даёт the normal verb to use for a plant "yielding" something?

Yes! The verb давать (to give) is the standard and most natural way in Russian to express that a plant, tree, or crop variety "yields" or "produces" a harvest (урожай).

Why didn't хороший урожай (a good harvest) change its ending, since it's the direct object of the verb?

It actually is in the accusative case! However, урожай is a masculine inanimate noun. For masculine inanimate nouns (and their adjectives), the accusative form looks exactly the same as the nominative dictionary form.