What is this?

It's called a "Turing test" after the English computer scientist Alan Turing who originated the test in 1950.

Why do we do this?

Simply put, because if we don't put some form of turing test in our web forms, we get inundated with form spam that is generated by automated tools run by spammers and hackers on the internet.

Why would they do that?

They may be "spamdexing" -- trying to get their website to show up on forums, bulletin boards or blogs on the net. If a web form posts directly to a web page without being examined, this is a way to get their website to show up in a LOT of places, which may help their ranking with search engines.
They may be trying to get website managers to buy their junk. Most website managers are too smart to buy stuff from people who resort to unscrupulous methods, but it only takes a small percentage of people to make it financially worthwhile to the spammers.
It may just be the hacker kid down the street who hasn't been taught better values than to abuse other people's resources. They think it's a cool thing to do, like bashing mailboxes.

The sad thing is, it's indicative that they feel so poorly about themselves that they believe legitimate endeavors are not sufficient to take care of themselves. They believe they have to resort to abusive behavior to get what they need. How sad!