One reason is that "being married" gives you legal rights that living together or being in a union do not. Sure we can extend many if not all these rights to homosexual couples without giving them the ability to marry but, and I'm going to try to do this without getting into the minutae of public international law, if you are married in one country the law considers you to be married in every country. So, for example, a couple married in Canada which moves to the U.S. can divorce in the U.S. or if one spouse dies, can take advantage of American law concerning married survivors under American estate law. They could not do that if they weren't legally married.
The second reason that homosexuals want the right to marry is that they are sick and tired of being treated as different than us heteros. And I see nothing wrong with them wanting to be treated as equals.