Europe and America stay in their respective corners, and nobody gets hurt.
The United States will stay out of Europe's business and leave their colonies alone. However, if Europe tries to invade America's personal bubble and add some Latin American colonies to their empires, the U.S. will consider that move an act of aggression towards their own country as well. And then it's on.
Because Latin American countries are totally independent now, and it would be, shall we say say, unfortunate if someone got in the way of that independence. (Cue steepled fingers and ominous laugh.)
The people of the United States were genuinely worried about more neighboring colonies coming under the rule of monarchies, because everyone knows governments spread like wildfire; that's why any loss of democracy down south was considered a threat.
The United States at this point is trying so hard to get into that cool club with all the world power, so Monroe bluffs to make America look like a force to be reckoned with.
After a few border conflicts and negotiations and several independence revolutions in Latin America, folks in the U.S. hear rumors that European countries have their eye on the Americas yet again. Britain tries to form an alliance with the U.S., but the president decides to go solo on this one.
The Monroe Doctrine, as it has become known, is only a few paragraphs in a larger speech. President James Monroe uses the framework of several recent events involving European nations and the Americas to lay out four main claims as a response (in this order):
The president begins this portion of his speech with a reminder about negotiations underway between the U.S. and Russia about the Russian Czar's claims on the Pacific Northwest. He also reiterates the United States' neutrality policy several times, particularly noting the recent Spanish civil war and French intervention as an example of how Europe doesn't have its stuff together.
The final big point that Monroe makes is his main argument for why this "doctrine" should exist. He says that you can't impose the style of government in Europe (a.k.a. monarchy) onto the independent republics over in the hip New World.
First of all, Europe is so far away; it's logistically challenging. Plus, it would be a threat to the U.S., a.k.a. the Original Independent Republic of the New World.
Anyone need a new band name? It's catchy and patriotic.
The United States has let messy, moldy-oldy Europe deal with its own problems, so Europe had better not try to take over anyone in the Western Hemisphere, because any new colonial nonsense will definitely be a threat to the U.S.