What does the phrase “Remember the Maine” mean?

Some other things to consider about that reference:

1)  It represents an example of "jingoism" - where extreme patriotism or nationalism is used to motivate a population towards a foreign policy goal, usually war.  "Remember the Lusitania" and "Remember Pearl Harbor" are other examples. (see link below)

2) At the time and for a long time after the Spanish were blamed for sinking the ship.  This was in no small part to the yellow journalism of William Randolph Hearst, newspaper magnate, who famously said: "You give me the pictures, and I'll give you a war". (Also linked)

3)  Recent historical and forensic research has led to the conclusion that the Maine sank when a coal fire ignited the gunpowder in her magazine, thus having nothing to do with the Spanish.  See the Random House article for more informaiton on this.