The inflamed flesh into which the nail is cutting is referred to as hypergranulation. Granulation occurs in order to heal wounds and is a perfectly normal process whereby excess tissue is formed to cover a wound until it heals. In the case of an ingrowing toenail, however, the nail is constantly cutting into the tissue, meaning it is not able to heal and thus the granulation of the skin continues. The image posted is an extreme case of this. I can also assure you that the smell is horrendous.

Edit: Just wanted to clarify that I speak from personal experience. I suffered a double hypergranulation, though not as bad as the original pic. I severely NOPED at the idea of surgery after having watched videos and decided to opt for conservative treatment which involved a stick of Silver Nitrate - a caustic substance which would burn away the excess flesh. Although I was told that granulated flesh wouldn't have a nerve structure, that was simply not true. Even after copious amounts of whiskey - which I had to have before starting the process - the pain was intense. I've since spoken to chiropractors about potential treatments and the best suggestion was to crush aspirin (anti-inflammatory) and mix with manuka honey, then apply. It should help with the swelling and pain.

TL;DR: Inflamed areas are due to hypergranulation. And speaking from personal experience, it sucks.