How does a mortgage-backed security issued by a bank allows it to borrow cheaper and make more loans?
Here are two ways by which selling mortgage-backed securities (MBS) benefited banks:


The MBS is less costly (or equivalently, more valuable) in the hands of the investor than in the hands of the bank.
The investor is misinformed/misled about the MBS and overpays for it.


Analogy for #1: When you buy car/fire/health/whatever insurance, you transfer the risk to another party (the insurance company) for whom that same risk is less costly (for example because they are able to pool/diversify risk across large numbers). This is the traditional, orthodox "markets are good" view. In this view, MBS's were a good development that further and more efficiently spread out risks and benefited everyone.

Analogy for #2: You suspect you have cancer and go buy some expensive health insurance. So you transfer this toxic asset (or risk) to another party (the insurance company) who, not knowing what you know, incorrectly judges the magnitude of this risk. This is to a large extent what actually happened.

(Note though that #1 and #2 are not mutually exclusive possibilities. Some of both probably did occur and continue to occur. But unfortunately, leading to the financial crisis, more of #2 and less of #1 occurred than was ideal.)