It's not explained outright in the movie, but the director kind of explains it in interviews prior to it's release (in response to the trailers, mostly.)
It's been known in the comics world for a long time that the adamantium metal in Wolverine's body is poisoning him. The only reason he can survive such a process is because his healing factor is constantly repairing the damage from the metal. In fact, there have been times in the comics when Logan loses his adamantium, and he is notably better off - faster, stronger, heals better, etc. The exact nature of this poisoning is somewhat inconsistent in the comics; in some places, the metal is called out as explicitly poisonous, but in other cases, it's explained as something specific to the bonding process Logan underwent).
In any case, in Logan, the metal is apparently poisonous. Logan says as much to Dr. Rice in the movie, and Dr. Rice does not even try to contradict him:
Logan: He's [Rice's father] the one that put this poison in me.
It's not clear if Stryker and his team knew this when they first applied it, but Zander Rice is at least aware of it's effects by 2029 (though he still chose to apply it to Lara).
Normally, Logan's healing factor keeps this problem in check. But as Logan gets old, his healing factor is starting to weaken. It's still working to keep him safe from the adamantium, but that prevents it from working effectively to heal his other wounds. He still heals, but slowly, and leaves scars. And if he suffers major traumatic injury, he may not be able to heal from it.
This is what the director had hinted at before the release of the movie, though not so specifically:
JAMES MANGOLD: 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.