How we cite our quotes: (Sentences numbered)
Quote #1
We shall endeavor in spite of this to keep the United States of America neutral. (2)
No matter what else happens, this is the top priority. The sky could be falling and Germany would be more scared of the U.S. entering the war against them. The Zimmermann Telegram is motivated by fear—this one very valid fear.
Quote #2
In the event of this not succeeding, we make Mexico a proposal of alliance on the following basis: make war together, make peace together…(3)
World War I started because nations made alliances with each other. Everybody promised to go to war for somebody else, so that if one country went to war they all did. Germany was worried that an alliance might also be the thing that would end the war. They wanted to get as many countries on their team as possible, especially if the U.S. was going to join the other side.
Quote #3
You will inform the President of the above most secretly as soon as the outbreak of war with the United States of America is certain and add the suggestion that he should, on his own initiative, invite Japan to immediate adherence and at the same time mediate between Japan and ourselves. (5)
You can tell that Germany is frightened of the U.S. getting into the war because they have not one, but two plans to try and stop it from happening. Plan one involves Mexico; plan two enlists help from Japan. That last part seems a little half-baked, but that's even more evidence that Germany is getting desperate out of fear.