I've heard this phrase used in economics. But it seems singularly inapt, the future is never here - how can you borrow something that doesn't exist?
Since you have asked only the "how" part, I'd explain here only the "how" part of the well-known phenomena.
That future isn't known implies that, (in economics) future is uncertain. Since future is uncertain no one knows if the expected future income will actually materialize or not when the future is actually here. Thus the one lending money against assumption of an having payment in the future carries a risk of non-payment. To compensate that risk the borrower agrees to pay "interest" to the lender.
That is how one is able to borrow from the future: by promising to pay extra in the future.
Thus it is just inexact language, for money is borrowed from the present only (in exchange of a promise to pay future money to lender). The borrower is borrowing money from lender who could had put money to some other use
On one hand it is a weakness of language.
You are basically taking from your future potential by spending it today usually at a premium cost. All things being equal, if you invested your resources instead of your future earning potential (via debt/risks) for money today, you would likely have more resources instead of having the resources immediately with additional risks attached to it. Therefore, loosely you are borrowing money today at the cost of even more money tomorrow.
On the other hand I disagree with the future not existing. A more accurate statement based on physics is that you haven't experienced it yet. Just because you have not yet experienced it does not mean it is not real since unless your immortal you didn't experience most of the universes past either.
@Madhur I agree, this is more of an economics question.