Learn Polish: Samochód miał być gotowy dzisiaj, ale znowu się zepsuł. - The car was supposed to be ready today, but it broke down again.

You're right that mieć means "to have". However, when you use it in the past tense followed by an infinitive (like być here), it expresses a past plan, expectation, or obligation. It's the most natural way to say "was supposed to" or "was meant to" in Polish.

Why do we use the masculine forms miał and gotowy for a car?

In Polish, samochód is a masculine noun. Because of this, both the past tense verb (miał) and the adjective describing it (gotowy) must take the masculine singular form to agree with it.

Why is there a się in znowu się zepsuł?

The base verb zepsuć means to break or ruin something else (for example, if you broke a toy). When a machine breaks down on its own, Polish uses the reflexive form zepsuć się (literally "to break itself").

Could I use psuł się instead of zepsuł się here?

Psuł się is the imperfective form, which would mean it "was breaking down" or "used to break down" repeatedly over time. Because the car completely broke down as a single completed event that ruined the plan, we need the perfective form zepsuł się.