edited for clarity. OSGi page gave a better simple answer than mine
A simple answer: An OSGi Service Platform provides a standardized, component-oriented computing environment for cooperating networked services. This architecture significantly reduces the overall complexity of building, maintaining and deploying applications.
The OSGi Service Platform provides the functions to change the composition dynamically on the device of a variety of networks, without requiring a restarts.
In a single application structure, say the Eclipse IDE, it's not a big deal to restart when you install a new plugin. Using the OSGi implementation completely, you should be able to add plugins at runtime, get the new functionality, but not have to restart eclipse at all.
Again, not a big deal for every day, small application use.
But, when you start to look at multi-computer, distributed application frameworks, that's where it starts to get interesting. When you have to have 100% uptime for critical systems, the capability to hotswap components or add new functionality at runtime is useful. Granted, there are capabilities for doing this now for the most part, but OSGi is trying to bundle everything into a nice little framework with common interfaces.
Does OSGi solve common problems, I'm not sure about that. I mean, it can, but the overhead may not be worth it for simpler problems. But it's something to consider when you are starting to deal with larger, networked, applications.