Is this fungus? What should I do about it?
This type of damage is very common in situations like yours. Both the wood and masonite are good sources of food for mold to grow if damp or wet for prolonged periods of time. Do not distress, most molds are not extremely dangerous. Since your area is small you can treat it yourself. Remove all obvious mold with a putty knife and wash the area with bleach. There is an anti-bacterial/mildew bleach available at most home improvemant stores that works well, but regular laundry bleach will do the job. Take the proper precautions, gloves, dust mask, eye protection, etc. to avoid getting hurt from the bleach or inhaling any live mold spores. Don't attempt to sand mold off as that can spread the spores, keep it confined. 

Let the cleaned area dry completely. Repeat the bleach wash if any traces of the mold is still visible after it dries. After the area is completely dried, you can paint the area to seal the boards. 

Remove any badly rotted wood back to the center of the closest floor joist. Replace the board. If you have only shallow damage on some boards, they may be fine after treated and dried as long as they are still solid and support a normal load.