Learn Turkish: Aslan pusudan önce kükreyemezdi. - The lion couldn't have roared before the ambush.

It combines the verb root kükre- (to roar), the inability marker -ye- (the -e ability suffix with a -y- buffer), the negative aorist -mez, and the past tense -di. The negative aorist past (-mezdi) expresses counterfactual past events ("wouldn't have"). Inserting the inability marker changes "wouldn't have roared" to "wouldn't have been able to roar" or "couldn't have roared."

What is the difference between kükreyemezdi and kükreyemedi?

Kükreyemedi uses the simple past tense, making it a factual statement about a specific past event: "It couldn't roar" (it tried and failed). Kükreyemezdi uses the aorist past, which makes it hypothetical or a logical deduction: "It couldn't have roared" (under those circumstances, it wouldn't have been physically or logically possible).

If I took out the -ye- and just wrote kükremezdi, how would the meaning change?

Kükremezdi is the standard negative aorist past, meaning "It wouldn't have roared" (counterfactual) or "It didn't use to roar" (past habit). The -ye- in kükreyemezdi is the crucial inability marker that adds the element of "capability", shifting the meaning from "wouldn't" to "couldn't".

Refresher: Why does önce require the ablative ending -dan in pusudan?