Spoilers everywhere. (Note that this answer will not make any sense at all to someone who has not seen the film yet.)
Aaron and Abe never definitively answer the question in the film. But there's a hint in the early scene where Aaron recoils his hand from the machine. He thinks he feels something, but isn't quite sure, saying "I don't know if I'm making it up". That's certainly a hint to the audience that neurological damage can result from exposure to these devices. Aaron and Abe themselves seem not to think about this later in the film.
The scene where they disconnect the battery and the machine keeps operating for some time gives a possible explanation of how the machine might disturb the nervous system. It indicates that in their experiment, electric current may have been sent backwards through time and thus "duplicated" by being subtracted from an alternate future that never happens, and added to their present.
Also don't forget that loss of handwriting isn't their only severe symptom. They are both seen bleeding from the ears at different times in the film. By the end, they don't even seem concerned about this.
By the end of the film, Aaron and Abe have endured several stresses that could contribute to their increasing health issues:
Prolonged exposure to the machines and the altered time-stream. The total amount of time they have spent inside the machines by the end of the film is not definite but it's an awful lot. Between the three time travelers that make it to the final scene I count at least:
- 5 work days plus a single 4-day-plus trip for Abe 1
- one work day plus a 2-day trip for Aaron 3 (who records the narration message for Aaron 2)
- Aaron 1 racks up a minimum of three 2-day trips, 20 work days, a 4-day trip, and multiple do-overs of the party. Even Aaron 3 can only speculate how many times Aaron 1 repeated the night of the party before he got it right.
All this in the span of what to everyone else was only three days. And this is not even counting all the Aaron's and Abe's who "get back in the box" or otherwise do not make it to the final scene.
- Sleep deprivation. Aaron and Abe seem to agree that sleeping inside the boxes is unnatural and is not as restorative as sleeping in their own bed would be. Abe seems to be also sleeping at home at night, but Aaron 3 is using at least part of the nights for his schemes. Both of them are clearly tired and irritable later in the film. It is possible that if both of them had stopped using the boxes for a prolonged rest period, the symptoms would have decreased or disappeared.
- While a bit esoteric, the fact that there are multiple permanent "copies" of Aaron and Abe at the end of the film opens the possibility of this unnatural situation being partly responsible somehow for their condition. Aaron and Abe themselves never seem to consider this. Aaron even cynically suggests that Abe use the machines to make copies of Aaron's wife and daughter to keep for himself.