Imagine you have two files, A and B, and you wish to update B to be the same as A. The obvious method is to copy A onto B.
Now imagine that the two files are on machines connected by a slow communications link, for example a dial up IP link. If A is large, copying A onto B will be slow. To make it faster you could compress A before sending it, but that will usually only gain a factor of 2 to 4.
Now assume that A and B are quite similar, perhaps both derived from the same original file. To really speed things up you would need to take advantage of this similarity. A common method is to send just the differences between A and B down the link and then use this list of differences to reconstruct the file.
The problem is that the normal methods for creating a set of differences between two files rely on being able to read both files. Thus they require that both files are available beforehand at one end of the link. If they are not both available on the same machine, these algorithms cannot be used (once you had copied the file over, you wouldn't need the differences). This is the problem that rsync addresses.
The rsync algorithm efficiently computes which parts of a source file match some part of an existing destination file. These parts need not be sent across the link; all that is needed is a reference to the part of the destination file. Only parts of the source file which are not matched in this way need to be sent verbatim. The receiver can then construct a copy of the source file using the references to parts of the existing destination file and the verbatim material.
Trivially, the data sent to the receiver can be compressed using any of a range of common compression algorithms, for further speed improvements.