Learn Turkish: Kuraklıktan ötürü sebzelerin büyümeyeceğini söylüyorlar. - They say that the vegetables will not grow because of the drought.

This is the core of a future complement clause. It breaks down like this: büyü (grow) + me (negative) + yeceğ (future suffix -yecek, with a softened 'k') + i (3rd person possessive, matching the vegetables) + n (buffer) + i (accusative case, because this whole clause is the direct object of söylüyorlar). Literally, the Turkish structure translates to "their future not-growing".

Why does sebzeler take the genitive suffix -in to become sebzelerin?

In Turkish complement clauses (when you say "that something will happen"), the subject of the action takes the genitive case. Because the action is treated like a possessed noun phrase ("their future not-growing"), "vegetables" becomes sebzelerin to link up with the possessive suffix -i on büyümeyeceğini.

I usually see için or yüzünden for "because of". What is ötürü?

Ötürü is another postposition that means "because of" or "due to". The tricky part is that it always requires the noun before it to be in the ablative case (-dan / -den / -tan / -ten). That is why "drought" (kuraklık) becomes kuraklıktan here.