its not backed by anything solid.. but it is backed by legislation..

EG the minimum wage laws make it so that a bank note is the equivelent to atmost X.X hours
for instance in california a $10 bank note has the belief that its back atmost 1 hour of sweat labour ($10 minimum wage)

yes i know people get paid more than $10 an hour.. hense why i said atmost..

also
tax that requires to be paid only in fiat is another reason that fiat holds value as it forces people to handle dollar. rather than choose a better currency to use.

inshort:
fiat is backed by legislation and the weak belief that it has value of sweat labour. but is not backed by anything of real value/substance.. which some say that paper bank notes are backed only by other pieces of paper (laws)