There's no toolkit per se; they most likely built a protocol on top of TCP/IP (i.e. on the application layer) to accomplish this, and implemented it in a program written for each OS. On each OS the program would access the system APIs to do things like intercept the mouse/keyboard and clipboard, and then send/receive that over the network just as any other browser or multiplayer game or network application would.
There have always been several ways before Dropbox to share files; for Windows computers, OS X has been able to access folders shared on a Windows network shares for a very long time. As for Windows, file sharing is a feature that's been around since Windows 95, as far as I know.
Other programs similar to this one include Synergy for Win/Mac/*nix, and Multiplicity and Mouse Without Borders for Windows.