Why did Germany officially acknowledge the contents of the Zimmerman telegram?

The German ambassador was acting prudently and addressing contingencies. Zimmerman was doing what ambassadors are supposed to do and what, no doubt, the British, Russian and French ambassadors were doing. The Zimmerman telegram merely proposed an alliance between Mexico and Germany IF the latter found itself in a war with the USA. Why shouldn't Germany have taken such possible precautions given the sham neutrality America practiced during 1914-17. When he admitted the telegram was his it was, no doubt, because how could it seem objectionable to seek alliances if the US declared war? It seems closer to the truth that Wilson was itching for war, and that the speed with which the Americans transformed the telegram into a pretext for it came as a surprise to the Germans.

If they'd denied it the US wouldn't have believed them anyway.

It would also have made them look weak. Who would go into a similar alliance with somebody who's prepared to rat on you like that?

Why did Germany officially acknowledge the contents of the Zimmerman telegram?

The United States could already verify independently that the British provided message was genuine even without Zimmerman's admission. Germany and the U.S. both understood that.

The way the Germans conveyed diplomatic messages to Mexico back in 1917 was encoding them and then giving them to the United States Embassy in Germany for retransmission. From the US Embassy they would go through a series of neutral countries before ending up in London to be transmitted across the ocean. British had cut Germany's transatlantic cables at the onset of the war and permitted the United States servicing Germany's diplomatic correspondence as part of President Wilson's peace initiative. Probable also because it allowed the British to intercept and read everything Germany was saying to it's embassies. From London the messages would go to the United States across a translation cable operated by the United States and Sweeden who were both neutrals, before being wired from the US to Mexico. So the United States already had the encrypted message directly from Germany. The United States could track it's origins all the way back to Berlin were it was hand delivered to the US Embassy. The British gave the US the translation along with the encrypted text, so it was already known that the message was unimpeachable. The United States had the encrypted text directly from Germany they just couldn't read it until the British did so for them.

Zimmerman Intercept (British Interception)
The message was delivered to the United States Embassy in Berlin and then transmitted by diplomatic cable first to Copenhagen and then to London for onward transmission over transatlantic cable to Washington.

Direct telegraph transmission of the telegram was not possible because the British had cut the German international cables at the outbreak of war.

In the United States the Zimmerman Letter was represented as Germany inciting Mexico to attack in exchange for money, arms, and territory. But that's a misrepresentation.

The Zimmerman letter was not sent to the Mexican Government (President Carranza) but to the German ambassador to Mexico. It was instruction to the German ambassador only to be pursued if the United States first declared war on Germany. As such Arthur Zimmerman's entire correspondence was being mischaracterized by the United States from his perspective. Zimmerman was astonished that the United States and President Wilson broke off diplomatic ties and ongoing negotiations with Germany after they received the Letter from Britain's foreign secretary Arthur Belfour.

These were the thoughts which Arthur Zimmerman presented in his March 29th, 1917 speech. That his letter was not a plot against the United States, but a contingency plan between two government functionaries sent by secure channels which was never to have seen the light of day if the unthinkable did not occur. The United States entering WWI. As it turns out it was used as the perfect excuse for the Wilson Administration to pursue a course of action which they were already strongly considering.

Prior to The United States being made aware of the Zimmerman Letter by the British. February 20, 1917 the United States was already well down the path of going to war with Germany.

• May 7, 1915, German submarine torpedoed the Lusitania, off the coast of Ireland. Killing 1,200 men, women, and children, including 128 Americans, lost their lives.
• July 21, 1915, President Wilson issues an ultimatum, to the effect that the US would regard any subsequent sinkings as "deliberately unfriendly".
• August 19, 1915 The White Star liner SS Arabic, outward bound for America, 50 mi (80 km) south of Kinsale. The loss of 44 passengers and crew, 3 of whom were American.
• Aug 28, 1915, German Chancellor issued new orders to submarine commanders and relayed them to Washington, stating that until further notice, all passenger ships could only be sunk after warning and the saving of passengers and crews.
• Jan 31, 1917 - Germany announces they are resuming unrestricted submarine warfare in the Atlantic.
• Feb. 3, 1917 - United States breaks off diplomatic ties to Germany.
• February 20, 1917 - The United States ambassador to Britain is presented with the Zimmerman Letter: see Use by the British Foreign Secretary Arthur Belfour.
• Feb 22, 1917 - Congress passed a $250 million arms appropriations bill intended to make the United States ready for war.
• February 28, 1917 - President Woodrow Wilson releases the Zimmerman Letter: see Use to the Press.
• March, 1917 - Germany sink four U.S. merchant ships
• March 29th, 1917 - Arthur Zimmerman gives a speech acknowledging his telegram to the German ambassador of Mexico.
• April 2, 1917 - President Wilson asks congress to declare war on Germany.

Zimmerman's actions reflected the style of his boss, Kaiser Wilhelm II. This was a blunt, heavy-handed "take no prisoners, make no apologies approach" that the Kaiser displayed in crisis after crisis; the Kruger Telegram congratulating the Boers on their suppression of the "Jameson Raid" that led to the Boer War; the "Hun statement" urging German soldiers to be merciless as Huns against Chinese "Boxers;" the Moroccan Crisis, that saw Germany trying to deprive France of Morocco; and the Daily Telegram affair that had Wilhelm listing France, Russia and Japan as his antagonists, (but excluding Britain).

You can characterize Zimmerman's actions as unwise, but no one can claim that he didn't speak for his government.

President Wilson revealed the existence of the telegram to the press on 28 February 1917. The press published the story the next day 1 March 1917 (and remember that Berlin is 6 hours ahead of Washington time). The German embassy in Washington would have had to send a telegram to Germany to inform them that the contents of what we now call "The Zimmermann Telegram" had been made public.

At that point, Germany had little choice but to respond, and to respond quickly.

Denial was never an option, since the cipher-text was still held in the telegraph company files in the United States and could be checked. The United States ambassador, Walter Hines Page, had these details (presumably from the head of Room 40, William Reginald Hall) and had included them with his report to President Wilson.

By claiming that his instructions (to the Mexican government)

... were only to be carried out after the US declared war ...

(quoted by @John Dallman in his answer above)

the German Foreign Secretary, Arthur Zimmermann, presumably hoped to minimise the damage that had already been done.