This has nothing to do with editing Wikipedia. The problem this demo is solving is "private information retrieval" -- that is, you send a query for a particular article, and the server sends back a response containing that data, but the server does not learn which article you asked for. Homomorphic encryption is just one of the building blocks used to achieve this.

A trivial solution to this problem would be for the client to just download a copy of the entire Wikipedia database and pick out the particular article it's interested in. With a proper implementation of PIR, the server still needs to scan through the entire encrypted dataset (this is unavoidable, otherwise its I/O patterns would leak information) but the amount of information that needs to be sent to the client is much smaller.