The reasons for owning books and DVD’s can vary.
As DataZak says, there’s the convenience factor of having your own copy. Yes, you could rent a given movie any time you want to watch it, and probably pay less through your lifetime than you would to buy it. But that’s also going to involve a trip out to get the movie, and what if you just want something comfortable to watch when you’re sick and there’s nothing on TV?
There are other reasons, too. I don’t know about you, but I am skeptical of any library’s ability to carry books I want to read. This is a carry-over from my days growing up in the 80’s in a town where the librarian was shocked and horrified when I asked if it were possible for the library to get a subscription to Analog. Their SF ran to (I counted one day) all of four books. One Asimov, one Clarke, and two Heinlein. None of which had been purchased after 1960. So I tend to buy books as a first choice rather than going to the library. I know this is an unfair judgement against my local library, now, but it’s still the way I think.
Movies are even worse for me. My tastes in film are such that most places that offer to rent movies won’t have the movies that I want to watch, even once. F’rinstance, the last time I was in Blockbuster, just after the US release of Nausicaa, they didn’t have a single copy. I made the decision to support anime translation by buying anything I might want to watch once, when USRenditions and AminEigo began in the 1990’s. And it’s worked wonders, too. Prices are down hugely, since. And I can even see some anime in Blockbuster these days. (Pokemon, alas, but some.)
My taste for ‘normal’ films is even harder to supply. My favorite Cary Grant films, for example aren’t things like North by Northwest, or An Affair to Remember, but I Was a Male War Bride, or People Will Talk, neither of which I’ve ever seen in any movie rental place.
With books there are additional reasons - owning reference works, for example, just makes sense. As an aspiring writer there are some topics I want to have the reference available 24/7, whether I’ve needed it in the past month, or not. Considering I tend to write things that involve fantastic of science fictional plots - my idea of a reference work can be pretty odd, too. (Now, what’s the lifecycle of the Komodo Dragon again? Do/did the Crow have neighbor tribes that revere Coyote as much as they despise him?)
And, frankly, there is simply the greed to it, too. Owning a library is something that warms the cockles of my heart. I grew up in a house full of books. My parents, when they bought their house, customized two rooms with each having a wall of bookcases. With more books still being stored in bedrooms and in the basement. Now that private ownership of video is approximately the same price as book ownership, a video library seems simply the logical extension of the same process.