A modern fax machine does not have the rotating drums and is a lot faster, but it uses the same basic mechanics to get the job done:
At the sending end, there is some sort of sensor to read the paper. Usually, a modern fax machine also has a paper-feed mechanism so that it is easy to send multi-page faxes.
There is some standard way to encode the white and black spots that the fax machine sees on the paper so that they can travel through a phone line.
At the receiving end, there is a mechanism that marks the paper with black dots.