Really depends. Initially it would be up to each person's own insurance policies to cover basic care. After that, would be a matter of whether or not state/federal government wants to step in and offer additional relief measures, or compensation. Red Cross and other charities often step in as well.
I know I saw on the news that President Obama had established a "relief fund" of sorts and was offering all of the assistance the United States had to offer. I agree with Sephira. The Red Cross has already been playing a vital role.
Ultimately we all pay for it through our charitable donations, higher insurance costs, higher consumer prices and higher taxes and/or fewer services. The massive explosion in Texas will also incur massive costs which will trickle down to all of us.
The city pays for damage to city property, tho they may carry insurance. Not sure how that works but most likely, in the long run, tax payers will pay for clean up and repair. Human injuries are paid for by the individuals insurance.