A "balance transfer" is when you use one credit card to pay off another.
Say you get credit card A with a 24% interest rate. You put $5,000 on this credit card. Later you get credit card B with an interest rate of only 12%. So you get company B to pay off what you owe on card A and put that amount on card B. Now you are paying only 12% on your $5,000 instead of 24%.
There is often a one-time fee to do this, the "balance transfer fee". In your case, 3% or $5, whichever is more. So in this example, you'd have to pay 3% of $5,000, or $150, to transfer the balance. But the interest is now 1% per month (12%/12), or $50, instead of 2% per month (24%/12), or $100. If you paid zero on the principle for 3 months, you'd break even. (Which you surely couldn't do because of minimum payment amounts, but I'm trying to make a simple example.) If it's going to take you a long time to pay off the debt, it might be worth paying the balance transfer fee.
Oh, and credit card companies often describe their rates in terms of "annual percentage rate". Most credit cards bill you monthly, so the interest you actually pay is the APR divided by 12 times your average daily balance for the month. And usually there's a "grace" clause that if you paid your balance in full last month and you pay in full this month, that you pay zero interest.