Short Answer 
The British recognized growing outrage in the U.S. against Germany in early 1917, prompting Admiral Hall to monitor public sentiment closely. He strategically released the Zimmermann Telegram on February 24, 1917, to incite American anger and facilitate entry into war, while withholding the fact that it was deciphered through British intelligence, enhancing its impact.
Step 1: Recognizing Growing Outrage in America
In late January 1917, the British were aware that public sentiment in the United States was shifting against Germany due to the announcement of unrestricted submarine warfare. This action was creating significant concern among the American populace, and Admiral William Hall recognized that this could lead to a potential U.S. declaration of war against Germany. Thus, he monitored the situation closely to gauge American reactions.
Step 2: Timing the Release of the Zimmermann Telegram
On February 24, 1917, Admiral Hall decided to strategically release the Zimmermann Telegram, believing it would provoke a stronger response from the United States. The telegram contained a proposal from German Foreign Secretary Arthur Zimmermann to convince Mexico to ally with Germany. Hall aimed to utilize this document to incite outrage in America and push the country closer to joining the Allied forces.
Step 3: Deception Through Disclosure
When Admiral Hall presented the Zimmermann Telegram to the American government, he opted not to disclose that the British had deciphered German codes to obtain the message. By withholding this crucial information, he enhanced the perceived credibility and urgency of the telegram, effectively manipulating American sentiment to favor entering the war alongside the Allies against Germany.