From the Apple documentation it just says:
When the value of this property is YES, the output file will be written in such a way that playback can start
When shouldOptimizeForNetworkUse is set to YES calling finishWriting will move the MP4 moov atom (movie atom) from the end of the file to the beginning of the file. The moov atom contains information about the movie file like timescale and duration. The moov also contains "subatoms" which contain information like the tracks, the data offsets in the file etc.
Playback is only possible using the information in the moov atom and when it is located in the beginning of the file rather than in the end of the file, a movie player can begin playing the file even if it has only loaded the beginning of it.
In practice modern browsers will fetch the moov atom from the end of the file using HTTP/1.1 Range header to load the moov atom even if it is at the end of the file.
Setting the shouldOptimizeForNetworkUse to YES also has a few side-effects: