From a first look on Windows, this is implemented with NTFS sparse files [1] and a file system minifilter. The sparse files act as placeholders so that you can browse your entire Dropbox structure. When you access a file, the minifilter sees that and starts fetching the data from Dropbox servers in the background. You can think of it as HSM [2]
[1] https://msdn.microsoft.com/en-us/library/windows/desktop/aa3...
[2] https://en.wikipedia.org/wiki/Hierarchical_storage_managemen...