I am looking to get my first mortgage and do not completely understand
them. My question here is this: Today's Prime Rate is 6.75% so how can
mortgage lenders set rates below that? For example, I found that the
US average for a 15 yr. fixed mortgage is 5.22%.

The prime rate isn't defined as the lowest rate you can get.  The Wall
Street Journal (WSJ) published "the prime rate" as the lowest rate
that 23 out of the 30 largest banks will offer (and I think it is in
terms of a 30 year fixed mortgage).

So their are 2 catches here:
1) 7 of the 30 banks offer a lower interest rate than the WSJs prime rate.
2) Banks offer other loans that might carry a lower interest rate than
a 30 year loan.

The example that you gave is a 15 year mortgage.  Currently, everyone
is expecting interest rates to go up, so long term fixed rate
mortgages are going to cost banks alot of potential profit in the
future (profit they could get by lending their money in the future at
higher interest rates than they can get today).  The 15 year loan
gives the bank more flexibility in the fact that they will get the
money back much sooner and therefore can loan it out again at the
higher interest rate.  So the bank is willing to lend for 15 years at
a lower rate than for 30 years.

A quick note... if you borrow $100,000:
30 year mortgage 6.75% = $649/month
15 year mortgage 5.22% = $802/month

Also be aware that you may not qualify for a bank's lowest interest rate.