I got a screen protector from the Apple store. I dropped my phone and the screen cracked. Isn’t the protector supposed to protect my screen????

Screen protectors primarily protect the surface against scratches. They don’t do much at all for a drop, especially if it lands on a corner or edge. A good protective case without a screen protector is more likely to prevent damage from drops.

Rudegar wrote:

when a material receive impact there are 2 responses absorbing or passing on the kinetic energy sort thick material can absorb but hard material which is usually required for transparency pass the kinetic energy on.

Tempered glass is kind of different. It can absorb shock (which will cause it to shatter) and take the brunt of the impact. It doesn’t protect from every impact as the OP found out.

I’d prefer a case however to reduce the chance of the screen breaking.

The protector may help but won't prevent damage in every situation.

See the following (as one example) from Belkin InvisiGlass Ultra Screen Protection for iPhone 8 / 7 - Apple

Up to 25% increase in drop performance, dramatically reducing screen cracking and breakage

Depends. They're not guaranteed to prevent cracking. I don't like them and I've seen them effectively create less clearance to the edge when a case is used.

when a material receive impact there are 2 responses absorbing or passing on the kinetic energy sort thick material can absorb but hard material which is usually required for transparency pass the kinetic energy on.