FDIC created what is called a bridge bank, which assumed all of SVB' assets and at least it's deposit liabilities (I'm not sure if it's credit liabilities were assumed, but I would think not). They have two years to wind down operations by either finding a buyer for selling off the business in pieces. If they cannot find a buyer, they will break the business into pieces and auction it off, either in terms of business units or it's underlying assets. All of those funds will go to depositors. Whatever liabilities remain will be covered by the Deposit Insurance Fund. That is an insurance fund run by the FDIC that all banks pay quarterly fees into to cover deposits in events like this. Interest of government bonds is also an additional source of funding. Coincidentally, the DIF is the same fund that SVB lobbied to not have to pay as much into.

Now, whether or not that means taxpayers are going to pay for it depends on your definition of a couple different things. First would be what you consider a bailout. Generally in the past, that term has mean that the government comes in to keep the company afloat, meaning the equity/shareholders retains value. That was not the case here. It also depends on if you want to say that customers are going to end up paying for the increased fees anyways because the banks will pass along those to customers. And, frankly, corporations pass along whatever expenses they think the market will bear, including taxes and fees. How much that's going to actually amount to on a per dollar basis for customers, I can't really say at the moment.