Basically, yes, the government just prints the money out of thin air, then charges us, the public for it (hence the nice, tidy term "bailout" (it's actually us bailing out these big corporations with our tax dollars)). That is why we are seeing inflation as high as it is. Prices of products aren't necessarily getting higher; it's the dollar's value that is getting weaker. Ever since the fed got rid of the gold/silver standard, the dollar's value has eroded.
And don't kid yourself that just because an account is FDIC insured, that the government will automatically give you you're money, if your bank fails. As long as banks fail one-by-one as they have been, the FDIC has been keeping their head above water, but if a lot of banks go under at the same time, we, the public, are screwed!