Learn Hardcore Italian: Al mercato compro spinaci e parmigiano per fare una torta salata. - At the market I buy spinach and parmesan to make a savory pie.

So al mercato literally means at the market or to the market, depending on context. In this sentence, it is understood as at the market.

Why is it al mercato and not nel mercato?

Both can exist, but they are not used in exactly the same way.

al mercato is the most natural way to say at the market / to the market in a general everyday sense.
nel mercato means in the market, emphasizing being physically inside it.

So in this sentence, Al mercato compro... sounds like At the market, I buy..., which is very natural.

Why is there no io before compro?

Italian often leaves out subject pronouns when they are already clear from the verb form.

compro already means I buy
So io compro is possible, but io is usually unnecessary unless you want emphasis or contrast.

For example:

Compro spinaci = I buy spinach
Io compro spinaci, lui compra pomodori = I buy spinach, he buys tomatoes

What form is compro?