Why does the ghost fall ill?

The answer depends upon your definition of "ill." In the story, the author describes the ghost as having fallen "ill" on two occasions.
The first occasion happens after the ghost tries to put on his customary suit of mail. Apparently, he hasn't worn it in a long time, and he has forgotten how thoroughly heavy and unwieldy it is. The result is that he becomes "completely overpowered by the weight of the huge breastplate and steel casque" and ends up scraping his knees horribly and "bruising the knuckles of his right hand."
The author relates that "For some days after this he (the ghost) was extremely ill, and hardly stirred out of his room at all, except to keep the blood-stain in proper repair." So, the ghost becomes indisposed after he injures himself while trying to put on the suit of mail.
The second occasion happens when the ghost tries to frighten the Otis family with his "Headless Earl" disguise. He hasn't worn the disguise in seventy years and thinks that he can do a tolerable job in frightening the Otis children with it. At the appointed time, the ghost rigs himself up in his outfit (complete with big leather riding-boots and a pistol) and makes his way to the twins' room.
When he gets to the room, he finds the door ajar. As he swings the door open, a jug of cold water comes crashing down on him, "wetting him to the skin." Because he is drenched with the cold water during the cooler hours of the night, the ghost catches cold; this is the reason that he falls ill a second time.

The shock to his nervous system was so great that he fled back to his room as hard as he could go, and the next day he was laid up with a severe cold. The only thing that at all consoled him in the whole affair was the fact that he had not brought his head with him, for, had he done so, the consequences might have been very serious.