Basically an advertisement for the newly-opened Hudson River Tubes — still in use by PATH trains today, over 100 years later — with the H&M lines proudly and boldly displayed in red. Planning for the future is also on display, making the service look somewhat bigger than it really was. From my limited research, it seems that the extensions shown in Manhattan were never actually built.
The Hudson Terminal Buildings (shown in the photo inset at top left) were replaced by the World Trade Center complex as part of the deal struck to allow the Port Authority of New York and New Jersey to take over the operations of the H&M RR.
Source: Penn State Maps Library/Flickr
I like how this map clearly shows why Christopher Street station was sited at Greenwich Street: for easy transfers to the Ninth Avenue Elevated! Today the El is no more, and transferring from Christopher Street to the West Side IRT requires walking 350 meters.
The Hudson Tunnels were also called the McAdoo Tunnels, named after William Gibbs McAdoo, who financed construction and led the efforts to link the two states by rail. McAdoo was touted as a serious candidate to run for the office of President in the 1920 US Presidential election, but political and personal issue (he was Woodrow Wilson’s brother-in-law) scuppered his campaign.