Seriously. The Zimmermann telegram seems like one of the most boneheaded decisions in the history of war. There was absolutely 0 chance it would have worked. Mexico knew it, the US knew it, and Germany should have known it too, so what was the point in sending such a damning letter?
Did they really think that with only having to deal with one front, they could stand against the millions of fresh troops and countless tons of new war machines we would bring to bear in a land war with them?
I know they were desperate to stop us sending supplies over, but how could making us a full on adversary in the conflict be a better option when they were 4 years into a fight that with their current enemies would end in a draw for them at best?
In hindsight the risk that the telegram would be decoded seems obvious. But at the time there would be no way to calculate such a risk. And keep in mind that in releasing the Zimmerman telegram British intelligence put a number of their own secrets at risk.
The telegram also had nothing to do with underestimating US military power. It was sent exactly because of concerns over the US getting involved. Mexico should be brought into the war if and only if the US was already joining in.
Ultimately the importance of the telegram has nothing to do with pure military strategy, but its effect on US public opinion. I would suspect that the German leadership would not be able to accurately predict this. They could not gauge how important public opinion would be for the US government's willingness and ability to declare war.