From the name, one might think it's made from fat. But apparently it's actually made from unflushable items like dental floss and facial wipes.
"Fatbergs (the word is a portmanteau of fat and iceberg) form over time as items that can't be broken down are flushed or washed down drains instead of disposed correctly"
So both oils/fat (which don't break down rapidly) and all sorts of things that shouldn't be flushed.
Those things collect fats and such, which bind them into larger assemblies. Two wrong things in the sewers that are worse together.