Times the rate divided by 12. It is no magic formula. The amortization schedule looks different because it is calculating the principal to pay the loan down over a certain time period. That changes as you reduce the loan.

The only way to lower your payment would be to recast your loan. Paying down on 6.25% is not a bad idea. But It should definitely be after other higher loans, 401k match, and probably tax advantage retirement accounts. After that depends on your risk tolerance. Over time there are better investments but 6.25% guaranteed is not terrible