He started aging, because he's old.
Circular logic, I know.
Speaking more to the scarring, director James Mangold explained in an interview, but it relates to his aging as well:
“One of the things we all thought about as we worked on this film is, well, we don’t want to rebuild everything. We want to have some questions. In order to make a different Logan, and a different tone of a Wolverine movie, we felt like we couldn’t hold on to every tradition established in all the movies religiously, or we’d be trapped by the decisions made before us. So we questioned whether Logan’s healing factor causes him to heal without even a scar. We imagined that it may have when he was younger, but with age, he’s getting older and ailing. Perhaps his healing factor no longer produces baby-soft skin. So we imagined he heals quickly, still, but it leaves a scar. The simple idea was that his body would start to get a little more ravaged with a kind of tattooing of past battles, lacerations that remain of previous conflicts.”
As Logan has gotten older, his healing factor has slowed down, causing him to age, and for his wounds to heal slower, resulting in scars.
It can be noted that Transigen has also been doing something affecting the water supply of the general public - it's unclear, but possible that they've added something to the water that has affected/poisoned the mutant population.