(someone's) balance

The "balance" of a bank account is the amount of money that you have in the account. So your account balance is a number. For example:

Your balance is $2,154.

The action most commonly associated with a bank balance is "checking":

When's the last time you checked the balance?

go with (something)

To "go with" something means to choose it. For example, you can say this when shopping for something like shoes:

I think I'll go with the red ones.

You can ask someone:

Which one are you going to go with?

And you can also talk about the past:

We went with the second suggestion.

a high-yield checking account

A "checking account" is a bank account that you can freely take money out of and put money into.

Usually you don't earn much interest from a checking account. So if you keep $10,000 in the bank for a year, you'll still have something like $10,000 to $10,025 at the end of they year.

However, a "high-yield checking account" pays a higher interest rate than usual. This allows you to earn more money. Usually there's some other restriction on this type of checking account. For example, you might need to maintain a minimum balance.

(someone) earns interest on (money)

"Interest" is a percentage that someone earns by lending money.

When you put your money in a bank, you're essentially lending the bank money. For some kinds of bank accounts, the bank promises to pay you interest on that money. For example, your savings account might pay 1% per year.

We say that you've "earned interest on" your money when you make money this way.