Exclusive checkout means reserving a file so that only you can edit it until you release it. It is the same idea as file locking; the phrasing comes from Perforce and older centralized systems, and it persists in game development vocabulary.
The word "checkout" causes real confusion for people arriving from Git, where checkout means switching which version of the tree you have locally and implies no exclusivity whatsoever. In the Perforce sense, checking out a file is an announcement to the whole team that this file is yours for now.
For binary game assets this is not a convenience, it is the only mechanism that works. Two people can edit the same Blueprint at once; nothing technical prevents it. What prevents the resulting lost work is that the second person is told the file is already taken before they start.
Unreal's editor has native support for the concept through its source control integration, which is why the Perforce workflow feels natural inside the editor. USourceControl implements the same semantics with server-authoritative locks: locks are enforced centrally rather than depending on every client behaving correctly.