2 main ways banks make money are by charging fees and giving out loans.

Checking accounts are accessed multiple times a day, so pretty volatile.

Savings accounts are accessed less (some even have limits, like 6 a month), so easier for a bank to use to give out loans. As a reward for allowing them to do that, they give you a cut of the interest they earn off loans. Bank of America and similar give terrible rates, starting at 0.01% APY, many online-only banks are now giving >3% APY, some even up to 5% APY (note that these will go down in the future once interest rates on loans go back down).

APR is the actual % that they use for calculations. However, sometimes they pay-out the earned interest multiple times a year, which means future interest is on the new, higher balance, so the pay-out is more, so for the year the actual amount of interest earned is more, this is APY, and this is the effect of compounding.

Examples:

1% APR paid once on $10000:
1% • $10000 = $100

1% APR paid semi-annually (twice) on $10000:
0.5% • $10000 = $50
0.5% • $10050 = $50.25
So an extra 25¢ where the APY is 1.0025%.

> Is your credit score looked at at all when opening one

It can be.

> Does having a better score mean you get a better rate on investment?

No.