British spies intercepted the telegram and gave it to the US government.Important telegrams were transmitted over long cables back then (they could be sent by radio, but then anyone with a radio could pick it up). Britain had cut Germany's cables to North America at the start of the war, so Germany asked to borrow the American cables; President Wilson agreed to allow it. But the American cable ran through part of Britain; British spies easily tapped into the cable and recorded everything that went through it- including Germany's telegrams. So the British caught the Zimmermann telegram when it was sent, and very quickly decoded it- the Germans had an elaborate code system that they thought was unbreakable, but the British had broken it early in the war.However, the British didn't want Germany to know that the codes were broken- as long as Germany didn't know, they would keep using the code and the British could keep reading their secret messages. So the British had to find another way to "prove" how they got the message without revealing the secret. They knew that the telegram would have to be transmitted through a certain office in Mexico, so they basically bribed someone in that office to give them a copy of the decoded telegram. Because the telegram had gone through American cables before being relayed to Mexico, the Americans would be able to tell that it was authentic.Once that was done, and the British were satisfied that the Germans still didn't know the codes were broken, the British gave the telegram to the American government; the Americans indeed verified that it had been transmitted, and President Wilson was furious.