They use source control! Source control keeps all the assets on a central server, and everyone can “get latest” of all the files onto their computer. After making a change, they can then push their changes back up to the server, which will store the new version (as well as the older versions, for safe-keeping and if you need to go back.)
Example source control systems popular in the game development business include Perforce (P4). The source control system will typically also “lock” things that others work on, so that you can’t modify them at the same time.
In general, different artists work on different assets. One character artist may own a particular character model. Then one or more texture artists own the textures that go into the model, and an animator may own the animations for this model. Before the “real” assets are done, stand-ins can be used that are replaced (and later updated) by the work done by the other team members.
Same thing for props and the like – individual buildings, rocks, trees, and such can be created separately in modeling tools.
When it comes to levels, it gets harder – two people can’t edit the same level at the same time. However, you can split the level into multiple sub-levels, and different people can edit different sub-levels at the same time. I think UE also supports layers, now, so two different people can edit two different layers of the same level.