A little simplified:
Private entities (banks and similar institutions) had lent money.
This had used up all (or most of) the money in the banks.
If they wanted to lend more, they would have to borrow it first in order to lend it.
They would then be (over-)burdened with debt.
The Fed(eral Reserve System) wanted the private entities to lend more money.
So, the Fed bought the original loans from the private entities.
This meant(1) that the private entitities were no longer concerned about these loans being repaid to them and (2) that they now had money (paid to them for the loans) which they could lend on, thus opening up the credit market.
In terms of pure logic, it would have been simpler for the Fed to lend money directly to people wanting to borrow, but that is a matter of economics and finance, not language. So, I'll now expand your original:
and buying up (from private entities) loans that private entities had made (to third parties) so (that) they (the private entities) could make new loans (to third parties) without being overburdened with debt.
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