Marat's disease has been speculated about at length, but we'll never know for certain. Though some of the more common diseases have been ruled out (syphillus, diabetes, eczema), the speculations that remain are specialized diseases at home among dermatology textbooks, but don't last in my mind long enough to remember their names. Suffice it to say it's not likely a common disease.

His skin condition is expected to have come on at some time during the early Revolution. I've seen estimates from 1788-1790 at the latest. It was a disease that made his skin very itchy and painfully burning, however it was alleviated by soaking almost constantly in a bathtub. Yes, Marat spent almost his entire day in the bathtub, and had a makeshift writing desk he used. He had several housekeepers (his housekeeper was supposedly leery about letting Charlotte Corday in, but Marat insisted since she said she had info on the Girondins.)

As for why the disease came on, the popular account is that it was either brought on or exacerbated by Marat's time hiding in the Parisian sewers. Marat had gotten word that a warrant had come out for his arrest, and he had been hidden in the sewers by his neighbours, who refused to give him up to the guard when they came to serve the arrest warrant. If the sewer incident had nothing to do with it, I've seen it stated that skin diseases that are genetic that don't appear until late adulthood are rare, so it was likely caused by something.