Basically, the US is in a major, major financial hole right now. Republicans (as a voting bloc, not as a group of politicians), as the party of small government, reduced spending, balancing the budget, etc, wouldn't go for a plan that would further complicate the already horrific financial hole that the USA is in. It would also go against Trump's "Drain the Swamp" message of reducing government waste, to propose a massive spending project. Therefore, he had to propose a way to build the wall without contributing to the debt/deficit (whether such funding method was feasible or not, by the way, was not a requirement, he simply had to say it; it is, after all, a campaign speech, and those are often based in not-quite-reality, from all sides)
FWIW (as an aside), it's extremely unlikely even he ever thought he could get Mexico to write a cheque to the contractors hired to construct the wall. I can't imagine anyone with any knowledge of how government finance actually works actually expected that. More likely, the US would apply tariffs, penalties, detrimental trade deals (which Mexico would have to take, because the alternative of not trading with the world's largest consumer economy would be even worse), and so on. From that, the US government would make money, and then Trump would request a spending bill from Congress of some proportionate amount to actually fund the wall. This is likely what "Mexico will pay for the wall" actually meant in real terms. As a non-American who doesn't follow national finance news that much, I don't know if he actually ever got those trade deals with Mexico in place to raise the money to fund the wall, and if he didn't, that complicates things quite a bit.