Could someone explain briefly why the higher the volatility, the lower the OAS for a callable bond? Thanks!

It has been explained over here several times! Use the search function buddy.

For a callable bond:

OAS = z-spread - option cost

Higher volatility increases the cost of the option, therefore OAS is lower.

You could also think that higher volatility increases interest rates, hence the OAS that you need to add to the curve to match the bond’s price will be lower, no?

You could think that, but you’d be wrong (or, at best, only half right).

Increased volatility increases the higher interest rates, but it decreases the lower interest rates. And the average rate stays about the same.