Why did Atticus ask Mr. Ewell to write his name?

Atticus had Mr. Ewell write his name to show that he was left-handed.

The girl was beaten by someone who lead with his/her left hand.

Tom Robinson could not even use his left hand, therefore he could not have beaten her; just proving further that the trail was a farce.

Atticus was trying to suggest that it was Mr. Ewell who beat Mayella, not Tom--Mr. Ewell was left-handed, and Tom couldn't even use his left-hand. Atticus then proceeds to question Mayella about her home life*, further driving his point home. Check out the link as well to get some more plot info.

*how her father is always drunk and angry and how she takes care of all her siblings on her own