Two questions that can disagree
Refinancing is usually presented as one question: does the payment go down? It is really two, and the answer to one is regularly the opposite of the answer to the other.
The first question is monthly. A lower rate, or the same debt spread over more months, reduces what leaves your account. The second is total. Spreading the same debt over more months means more months of interest, so a refinance that lowers the payment can easily raise what the loan costs in the end. Take a loan twenty-eight years into its life and refinance it into a fresh thirty-year term and the monthly saving is real, the total is worse, and both facts matter.
This page always shows both. The headline is the break-even month, because that is what people come for, but the lifetime figure sits directly underneath and says plainly when it points the other way.
How the break-even month is worked out
Closing costs are paid in cash at signing here, not added to the balance. That is a convention, and it is the one that makes a break-even month mean anything: roll the costs into the loan and the two loans are different sizes, so there is nothing clean left to compare.
The break-even month is then simply the first month at which the accumulated monthly saving covers those costs. Save two hundred a month against four thousand in costs and you are even in month twenty. If you expect to sell or refinance again before that month arrives, the deal costs you money however good the rate looks.
When the new payment is higher, there is no break-even month and the page says so rather than printing a negative number. That case is not a failure: shortening the term almost always raises the payment, and what you get for it is a much smaller interest bill.
Why we do not ask for your current payment
The form asks for the balance, the rate and the months left, and derives the current payment from those three. Asking for the payment as well invites a set of four figures that contradict one another, and a calculator that silently believes the wrong one produces an answer that looks authoritative and is not.
The derived payment is shown under the field so you can check it against your statement. If it is noticeably out, one of the three inputs is wrong, and usually it is the months left: count from the payoff date on your statement, not from the original term.