Harry goes to the staff room, hoping to get back the library copy of Quidditch Through the Ages Snape had confiscated earlier that day. The only people in the staff room are Snape and Filch. Filch is helping Snape to bandage a large wound on his leg that has been causing him to limp. What Harry hears Snape say about sustaining the injury makes it clear he was bitten by Fluffy (PS11).
This chapter begins the first week of November. Harry's run-ins with Snape happen on Friday the 8th, and the Quidditch match is on Saturday the 9th. (There is a remote possibility that the events are taking place the next week Friday and Saturday, November 15 and 16, but it is not very likely.)
Why didn't Snape go to Madam Pomfrey for help with his wounds? She does have a reputation for confidentiality when it comes to strange wounds; she never does tell on Ron when he shows up with a dragon bite, for example. Why Filch? Snape and Filch do seem to be on fairly friendly terms in this book.
It is possible that the bandages or other materials have magical properties, but Filch cannot do magic. It appears as though they are using non-magical means to treat the bite.