Why is the point where price equals average variable cost considered the shut down point?

This is because once the price goes below that point, a firm is better off just closing down.

Let's think about what happens if a firm is producing at a price that is below their average variable costs.  Let's take a movie theater, for example.

Let's say it costs a theater $5 per person in variable costs to show a movie -- counting costs like the ticket takers, the electricity, the janitors, etc.  If the theater can only charge $4.50, for some reason, it loses money every time someone comes through the door.  So it should shut down.

If price is higher than AVC, but below average total costs, it should stay open because even if it loses money, it would still lose MORE money by closing.  That's why the ATC point is not the shut down point -- you still if price is less than ATC but more than AVC, you still lose less money by staying open.