With history, everything is interconnected. Kind of like seasons of American Horror Story: it may feel like you've moved on to a completely different era with different politicians and problems, but then you realize that the same actress who was a ghost last season is now a circus freak.
And also that war that happened decades ago is still totally a thing.
What we're getting at: to understand the Zimmermann Telegram, you have to know something about the Mexican-American War. As its title suggests, it was indeed a war between Mexico and the United States. So far so good.
In the mid-1800s, Texas had been allowed to become a U.S. state without anyone being clear on just where Texas began and Mexico ended. Really, they weren't in agreement as to which river was the U.S.-Mexico border. So almost immediately after joining the union, Texas started a war with Mexico giving the rest of the U.S. some serious buyer's remorse. But at the time, warranties on defective states were notoriously complex, and who really wants to stand in line at the exchange counter all day?
So the U.S. had to keep Texas and just try to make the best of a difficult situation.
Actually, the U.S. used this war to their advantage in an impressive, if somewhat dishonest, way. They dominated the Mexican military, and when Mexico surrendered, the U.S. used the treaty to "clarify" the Texas border, which actually meant taking about one-third of Mexico for themselves.
What started out as just a skirmish over whether or not some empty desert land was in Mexico or in Texas wound up giving the U.S. the land that would become California, Arizona, Nevada, Utah, and parts of Texas, Oklahoma, Kansas, New Mexico, Colorado, and Wyoming. It was an embarrassingly one-sided victory that created a significant portion of the United States. And even back then, in a time of rampant racism and unhinged imperialism, there were those who thought that the U.S. basically stealing a sizable chunk of a continent away from their impoverished neighbor was a bit mean-spirited and bullying.
It's safe to say that Mexico didn't feel very good about this deal either. And those hurt feelings are what German Foreign Secretary Arthur Zimmermann was attempting to tap into in his Telegram. He was suggesting that Mexico try to win back some territory by attacking the United States. He was trying to kick off a Mexican-American War 2.0 by telling Mexico to go get what was stolen from them in (what he hoped) was just the first round in an ongoing North American brawl.
Actually, Germany didn't really care who attacked the United States—as long as somebody did. Well, somebody in addition to Germany, because Germany would certainly be attacking U.S. ships in just a few weeks due to their new policy of unrestricted submarine warfare. Mexico just seemed like the most likely candidate: they were geographically close to the U.S., they had reason to hate the U.S., and they had lost territory to the U.S.
Germany had a history of attacking countries for these exact reasons, so by their logic, this scenario was a no-brainer.
Germany just really wanted a diversion. They wanted to blockade Britain and France (as part of that whole World War they'd gotten themselves into), which meant no ships, not even ones from neutral countries like the United States. But they also knew that Americans would get all weepy and shout-y when Germany's submarines started sinking their ships, especially ones filled with women and children. But if the U.S. was also being invaded from the south (and maybe the west, too, if Japan was game), the Americans might not notice when some of their loved ones ended up on the bottom of the Atlantic.