The vehicles provide cash in the form of (hopefully low interest) loans specifically for device/equipment purchases and network buildout. Since more cash goes out of Sprint and than comes in, the threat was that the company would literally run out of cash to keep the lights on. To get more cash, Sprint might have needed to issue more high-interest company bonds (debt) that my buyers may not want or issue more shares (at really low prices diluting current shareholders). Running out of cash is accelerated my the new leasing and Easy Pay plans that force Sprint to buy full-price devices up front and then wait for customers to pay for it in installments.
The SoftBank-engineered vehicles will in theory handle all those cash transactions (such as upfront payment for devices), thus leaving Sprint still being able to pay its other day-to-day bills (such as employee paychecks! and interest payments on older debt from banks, etc.) with cash while adding more "paper" debt to Sprint's balance sheet.