On the simplest level, a signature is simply a way to make someone legally responsible for the content of the form. But in addition to the legal aspect, the signature is an appeal to personal integrity, forcing people to consider whether they're comfortable attaching their identity to something that may not be completely true.
Turns out that we're more likely to report the complete truth if we contemplate our integrity before filling out the document instead of after. It's a wonder no one has thought of this before now.