Commercial banks do not "get money" from the government. They do so either from the FED or either from worldwide financial markets. And actually, banks can even buy gov's bonds, which in this case is the contrary of what you describe.
Conceptually, the FED simply buys bonds emitted by commercial banks. But using the word "buy" here is misleading. The FED is not a traditional economic agent.
First, note that the FED has two main possibilities to lower interest rates. They can do it directly by saying, "ok guys, tomorrow refinancing will be less expansive" or indirectly by injecting liquidity in the system, i.e. by proactively buying commercial banks' bonds for (almost or really) nothing, or even paying them interests for that -- which is equivalent to having negative interest rates!
When the FED does so (either directly or indirectly), commercial banks will be able to refinance themselves at a lower cost. Which means that at the end, these commercial banks will also be more, say, flexible, and thus will stimulate demands (here and there) in the national economy by lending money (here and there). If supplies stay constant, markets will reach a new equilibrium via prices rising due to these stimulated demands $\iff$ stops deflation.