When I buy a game, what do I actually pay for?

When you purchase a game from the store, you tangibly receive a specialized piece of plastic with software on it, and some packaging. Everything else is intangible: rights, a license, guarantees, etc.

Typically, and I can't stress that word enough, what you get in layman's terms is a license to use the software on the disc for personal use in the privacy of your own home and a guarantee that the hardware disc is free from defects in workmanship. You do not receive any copyright rights over the content, nor does the license permit you to duplicate or redistribute the content (well, it does to some extent, usually for backup and for actually installing and running the software).

When you record yourself playing a game and then post it on YouTube what you're doing is duplicating a portion of the audiovisual intellectual property in the software and redistributing it in a different form. That's why it's a copyright violation, and that is why the license you are granted as an end user doesn't permit it.

Don't I "own" the content I paid for?

No.

Or have I only paid for the service i.e. the ability to be able to play the game and only that?

Yes.

Though you probably can "quote" portions of the game for reviews etc. under fair use laws (which may or may not be applicable in your country). The amount (i.e. length of the video) will be relevant here. A 2 minute clip might be OK, but a 15 minute one without commentary might not be.