It's important to put this into historical context. During the crises, we were in essence suffering a bank run. Liquidity crisis is code for bank run.
But modern bank runs look different from old time bank runs. Modern banks are divided into (roughly) two camps. Local depositor banks and larger investment banks. There is a complex inter-bank lending market that connects the different types and assures banks have enough reserves to avoid insolvency. In a modern run, banks actually fear losing the support of other banks more so than depositors.
This is important as all banks are inherently insolvent and need to be propped up by other banks in turn by a central monetary source (like the Fed). This is because they mismatch short term liabilities with long term assets. If I take out a one year loan to buy a ten year bond, I am very dependent on that one year being perpetually rolled over. Same with bankers. The Fed subsidizes the short term debt market to them, so they can make more money mismatching short term debt with long term assets.
During the crises, the money market shut down. To academia...this wasn't supposed to happen because of the open market. But it did. The reason it did was because banks stopped trusting each other and we had a very modern bank crisis on our hands. Banks tried to reverse maturity mismatching by holder more short term assets (like reserves), but this is impossible in the aggregate and actually causes more of a crisis.
The idea behind the Fed paying banks for reserves, is that it provides them with a source of short term assets to help deal with pressing short term liabilities. QE on the other hand helped deal with banks change long term assets for shot term from the Fed.
Normally, the Fed bails out (and does so constantly each year) bank crises through the open market. If a bank is seeing their short term debt called in, they borrow from the money market which in turn borrows from the Fed. So indirectly banks are guaranteed their source of short term debt to perpetuate their maturity mismatching.
Whether the government should be using public resources to prop up such behaviour is a very good question. Some central banks actually have considered measures to CHARGE interest on excess reserves, to encourage lending on the Fed Funds market and to help other banks overcome their liquidity concerns.