Why did the Zimmerman telegram upset America if it was not asking for aggression against America?

I recommend you read up on the entire situation. The Zimmerman Telegram from Barbara Tuchman is an excellent start.

There certainly was a kind of aggression against America: promising several US states to Mexico is meddling around in forbidden political territory. Compare this with the US hypothetically promising Schleswig-Holstein to Denmark. "In case of war" wouldn't be good enough for Germany. One does not promise parts of countries to other nations, no matter what. Doing so is an act of war.

That's one pretty firm reason. Another is the unrestricted U-Boat warfare the German navy was going to introduce. That would directly affect US trade relations and economy.

The announcement when unrestricted warfare at sea would continue came a few hours before it started. It wasn't as close as the Japanese declaration of war after Pearl Harbor, but not by much. That also is highly aggressive: neutral nations didn't get any chance to warn their ships.

There was a very strong peace / let's keep out of the war movement. Wilson was firmly in that camp. He did everything possible, and a lot that wasn't even legal, to keep out of the war.

The Zimmerman Telegram was the straw that broke the camel's back. War would have happened eventually, simply because of the unrestricted U-Boat warfare. Only much later. Germany expected to have won the battle of the Atlantic by then.

States bordering the Atlantic were in favor of war for a long time already. Western States bordering the Pacific were more worried about a Japanese invasion, possibly from Mexico. Central states didn't care. That effectively prevented American entry into the war. (ref: this comes straight from the book of Barbara Tuchman.)

What the telegram did was to unite the country. Within days, the peace movements were gone. Before the war, Germany boasted they could mobilize millions of German Americans. A lot of (German) Americans felt some sympathy for Germany. That evaporated overnight.

Mob mentality: they are scheming against us!!

Yes, the telegram was talking about situation "in case of war between USA and Germany":

We intend to begin on the first of February unrestricted submarine warfare. We shall endeavor in spite of this to keep the United States of America neutral. In the event of this not succeeding, we make Mexico a proposal of alliance on the following basis: make war together, make peace together, generous financial support and an understanding on our part that Mexico is to reconquer the lost territory in Texas, New Mexico, and Arizona.[...]
— Zimmernam Telegram on Wikipedia

But to an average reader it means "we are going to sink your ships, and we are banding together with your enemy to take your land".

The relationships between the USA and Mexico were bad, the sentiment against Germany was also relatively high. Now, the public opinion had a proof that those two "almost enemies" are ready to join forces against Americans. The telegram wasn't exactly the casus beli, but it has managed to swing opinion further towards the war against Germany.

Like with most scandals with legs, a large part of the Zimmerman scandal in the US wasn't the details of the initial scandal, but how badly it was handled afterwards.

As this question points out, it seems like it would have been more sensible to deny the memo publicly, or perhaps publicly fire the author, disavow it, and apologize. That's what good diplomats would do. And in fact, that's what their lower-level professionals and allies in the US press immediately set out to do.

At first, some sectors of the US papers, especially those of the Hearst press empire, questioned whether the telegram was a forgery made by British intelligence in an attempt to persuade the US government to enter the war on Britain's side. This opinion was reinforced by German and Mexican diplomats, as well as pro-German and pacifist opinion-formers in the United States.

Unfortunately for Germany, the country was run by Kaiser Wilhelm, who had the typical personality of an authoritarian politician. He was obsessed with his status, could never be wrong, and had the motivational instincts of a bully. Anyone who objected to the country running its domestic or foreign policies along those lines got sacked, starting with Bismarck (arguably the greatest European politician of his age).

So what they ultimately did instead of any of those sensible professional diplomatic moves, was to deny that what they did was any kind of problem whatsoever, and accuse US officials (and the public that popularly supported them) of being babies about it.

However, on 29 March 1917, Zimmermann gave a speech to the Reichstag confirming the text of the telegram and so put an end to all speculation as to its authenticity. By that time a number of US ships had been torpedoed with heavy loss of life.

...

He also said that despite the submarine offensive, he had hoped that the USA would remain neutral. His instructions (to the Mexican government) were only to be carried out after the US declared war, and he believed his instructions to be "absolutely loyal as regards the US". In fact, he blamed President Wilson for breaking off relations with Germany "with extraordinary roughness" after the telegram was received

This is of course the exact kind of delusional egotiscial bullying that had landed them at war with nearly all of Europe on multiple fronts at once. Denying reality may sometimes work to steamroll local political opposition, but when it hits outside reality the results typically aren't pretty.