On Illegal Immigrants and Voter Fraud

I've never understood the allegation that illegal immigrants are trying to vote to influence elections. Why would an "illegal immigrant" risk losing everything by ever going near a polling place, some of which are located at police departments and official government offices.

Such people spend their entire lives hiding in the shadows. Their whole existence is predicated on the fear of discovery, capture and deportation, yet we are to believe they are attempting to vote fraudulently -- and in sufficient numbers -- so as to sway an election?

Such claims defy all logic and reason.