The major reason why states require people to register to vote is so that they can be sure that the people who are voting are actually eligible to do so.
If people could just come down to the polling place on election day, it would be awfully hard to make sure that they were really eligible to vote. There would be no time to check on their addresses or anything like that. This would lead to all sorts of opportunities for fraud and all sorts of allegations of fraud from the losing side.
So it is important to have voters register so that we can be relatively certain that only eligible voters actually vote.
Your question asks about the importance of registering to vote, and the answer is that it's imperative. In order to take advantage of one's right to vote (as well as to perform one's civic duty), one must first register. No registering, no voting. It's clear that encouraging people to vote has become a priority in the last decade or so, as evidenced by the ease with which one can now register to vote. When I recently moved to another state, for example, I signed up for my voter registration card when I got my new driver's license; the card came in the mail and I'm ready to vote next week. College campuses often conduct voter registration campaigns, especially since many students on campus will vote for the first time while there.
As we approach elections this year, the drive to register voters is surging. No registration, no vote. No vote, no voice. No voice, no right to complain.