Correct spelling, explanation: the word worsened is a past simple/past participle form of the verb worsen. As you can see, it’s a regular verb, which means that its conjugations are created by adding the -ed suffix to the root. Derivatives must always accord with word-formation bases. That’s why the only correct form is worsened, not worstened.

Definition of worsened:
verb (past simple/past participle), make or become worse
My mother’s condition worsened when we made it to the hospital.
I think that my health has worsened over the years.

Collocations with worsened:
Some most commonly used collocations include: worsen with age, worsened by stress, and worsen progressively.
My friend was diagnosed with ADHD when he was a child. Unfortunately, his symptoms have worsened with age.
Mark’s mother has a serious heart disease that can be worsened by stress.
The child’s stomachache worsened progressively, so the parents must’ve gone to the hospital.

Incorrect spelling, explanation: the word worstened doesn’t exist in any dictionary, but this mistake is easy to understand. It’s based on the assumption that the basis of this verb is a superlative degree of the adjective bad – worst. In fact, the basis is a comparative degree (worse), which can be logically explained. Worsened is the only correct form.