Checking a Curator Production out
In Curator, when you check out a Production, it means the Curator for Adobe panel downloads the .prproj that is saved against the Production asset and opens it on your local machine. The media inside the project file is not downloaded, as Curator works with streaming proxies (which are securely loaded when the project is checked out). This means that it's much faster to load the Production with streaming proxies in it, rather than consolidated media items.
Checking a Curator Production in
In Curator, checking in a Production means you save the version of the .prproj you have open in front of you back into Curator (you may have made some changes to it after you checked it out). It will be saved as a new version which means you can choose which version you check out next time you want to work on it.
Note that only one user at a time can check out the same production. This is to ensure that versioning functions correctly and other users are not overwriting each other's work so be sure to check your production back in once you have finished your edits!