How can a loan be permissible when the Prophet, may Allah's prayers and peace be upon him, warned against it?

The ruling on borrowing for the borrower is permissibility due to need. The Prophet, peace and blessings be upon him, himself did this. One who intends to repay a debt but is unable to do so is not blamed شرعًا (religiously/legally), and Allah will take care of its repayment, for the Prophet, peace and blessings be upon him, said: "Whoever takes people's money with the intention of repaying it, Allah will repay it on his behalf."

As for discouraging debt, this applies to one who deliberately betrays or takes it without need until he becomes unable to repay it. Lending is recommended for the lender; every loan is a charity, and whoever eases the burden on one in hardship, Allah will ease his burden. As for discouraging the borrower from taking a loan when there is no need, it is due to the Prophet's saying, peace and blessings be upon him: "The soul of a believer remains suspended by his debt until it is paid off."