There are lots of different reasons for various organisations to move to DevOps.
I'll try to list the ones that come up often.
Reduce time to change cycle
There is often a long time between making a request for change and it actually being deployed and used in the organisation. First it is planned in one of the development cycles by the developers and after it is delivered it is planned in one of the release cycles of operations. Both cycles include testing and in case of problems found, both cycles reset. By integrating development and operations departments, we can streamline both processes.
Software vs Hardware issues
Remember the Bugs Bunny cartoon where Bugs and Daffy are arguing whether it is duck season or rabbit season? Now imagine we instead made it with developers and operations where developers argue it is a hardware issue and operations argue it is a software issue. To the end user this is a distinction without difference. They just want it fixed.
By bringing developers and operations together they'll have to fix the issues. And it may turn out it was a software and hardware issue.
Us vs. Them
In a lot of companies the distance between testers and developers was growing because they were seperate departments and the development cycle was getting more and more formalised and standardised.
With the coming of Agile, developers and testers have been working closer together and we've started seeing each other's point of view on the development cycle and maybe even come to respect it.
Something similar needs to happen between developers and operations, because as both fields mature and processes further formalise and standardise, the distance between these departments is growing.
So one of the issues with the traditional model is that it seems like "us" vs. "them" for developers and operations alike. Both not completely understanding the difficulty of the other's responsibilities.
With DevOps both specialties will learn some of the skills traditionally performed by the other. Noone will expect a system administrator to become a software engineer or a developer to become a network engineer, but both are expected to take on some of the other's responsibilities. This means that when you really need some extra hands, they are there.
And there are some definite upsides for developers: you now have more control over your test environments, you will find it easier to have software deployed to the users and have more people in your organisation to share your love of the craft with.