You've got to understand three things if what I'm about to say seems bizarre, as it naturally would to most Americans:
The idea of a leader who wasn't the complete and total master of "his house" was something that many Russians from Nixon's generation, even educated ones who knew things about the world outside of the USSR, would have had a very hard time processing. Having cut your teeth under Josef Stalin leaves an imprint.
These men lived in a country where far graver violations of civil liberties than Nixon ever dreamed of were still very much a daily, routine occurrence in the 70s, and had internal politics that could be charitably described as Byzantine and completely cut-throat. There's no political scene on Earth that makes one more justifiably prone to conspiracy thinking than Russia's, then and now.
The Soviets saw Nixon and Kissinger as the good guys, as cold, realist colleagues of a sort, in spite of their hawkish, unambiguous anti-Communism and their courting of China. The Soviets wanted a business-like, stable relationship with the US with no messy factors like human rights in the USSR involved and that focused on practical matters at hand, like not sending the world into nuclear Armageddon or dividing up spheres of interest. Nixon was all about that. Some of the most ardent foes of Nixon during Watergate were all about preventing that, both on the left and the right, and it showed: it was a combination of the two, not one or the other, that used Watergate to interfere with the still fragile process of detente. Liberals wanted to focus on human rights rather than arms control, and in general didn't want Nixon to use a foreign policy success to get away from Watergate, conservatives (in both parties) were increasingly opposed to arms control with the USSR in principle. The fact that this was in the context of vast changes in the Middle East also evoked the specter of the Israel lobby in Moscow's eyes.
So, given all this, here are a couple of passages that will illustrate what the Soviets thought of Watergate:
"To the Soviet leadership such a precipitous collapse after Nixon's visit to Moscow still came as an unpleasant surprise. While such an outcome was evident from the course of events, there as perplexity in the minds of the Kremlin leaders, who were at a loss to understand the mechanics of how a powerful president could be forced into resignation by public pressure and an intricate judicial procedure based on the American Constitution-all because of what they saw as a minor breach of conduct. Soviet history knew no parallel... But many Soviet leaders including Brezhnev believed that the scandal was being used against Nixon by opponents of detente. Nixon's conversation and message opened a unique personal exchange with Brezhnev that has never before been chronicled. As Watergate turned against the president, Brezhnev proved himself Nixon's staunch friend and supporter, probably the last he had among the leaders of great nations, including his own."-Anatoly Dobrynin, Soviet ambassador to Washington.
"In 1974, I spent several months in the dusty main reading room of Moscow’s Lenin Library, studying the troubled Soviet-German relationship. It was a thrill to glimpse the closed world of the Soviet Union, yet I soon found myself spending lots of time explaining the United States to incredulous Soviet students at Moscow State University. Yes, I said, Americans were outraged that President Richard Nixon’s White House was listening to people’s phone conversations. And no, The Washington Post was not in the hands of a capitalist-Zionist cabal out to destroy detente by fabricating Watergate. On the day Nixon resigned, I left Moscow, having failed to convince my fellow students about how America really works."-Angela Stent.
Many Russians, including some of the top people in the Soviet government, couldn't believe that Nixon was really taken down over what one Soviet Politburo member referred to as "stealing some silly documents." Beyond that, they viewed many of Nixon's domestic political opponents with disgust, as they knew that many of them would change the course of policy not to their liking. In one ironic twist, Leonid Brezhnev remained staunchly sympathetic to Nixon long after everybody in Washington stopped doing so in the later stages of Watergate, sending several telegrams of encouragement to him. In contrast to what many expected, Brezhnev never attempted to use Watergate for his political advantage. Wiretapping and harassing political opponents was considered to be an entirely expected, natural thing for leaders to do, from the POV of the Kremlin. (Dobrynin: "Who cared if it were a breach of the Constitution?") They couldn't believe that the Americans would "destroy their own government" over such "petty nonsense". Many in Moscow, as a result of all of this, saw darker forces at work in Washington and perceived Watergate to be some sort of coup or power struggle. Some even had a hard time grasping that Nixon didn't control the media.
As for the effect this had on history, just as Washington had pro and anti-detente factions that changed over time, so did Moscow. Many in Moscow-and in the KGB in particular-did take this as a sign of weakness. There were people who questioned whether it was worth accommodating a country that could be this "stupid" (for those in the KGB who did understand that, yes, Nixon fell because of that) and assured skeptics in the Politburo that the US was fundamentally too self-absorbed and weak to seriously pose an obstacle to Soviet adventurism. Which in fairness, at the time, they would have been exactly right. But it was a relatively short term crisis of confidence, whereas Soviet economic problems were a lot more systematic and long-term. Soviet officials from the period, including Dobrynin, have stated that once Nixon was gone and the Ford Administration was facing immense problems at home and abroad, they took far less seriously American objections to Soviet adventures in the Third World. (For example, note that the Soviets did little to stop Nixon and Kissinger's covert actions in Chile. Andropov noted in one internal memorandum that, as Nixon and Kissinger had tacitly accepted that Poland and Czechoslovakia were areas of critical Soviet interest and that the USSR was free to act as it pleased, so was Latin America for the USA and Soviet policy had to be one of caution) It wasn't for nothing that Nixon's resignation coincided with the beginning of a more aggressive Soviet foreign policy in the Third World. Weakness, real or simply the perception of it, has always triggered a response in Moscow.
Watergate and other unfortunate events at the time, such as Brandt's resignation and the flare up in the Middle East, did not kill detente all by itself-thankfully, we still went on to sign the Helsinki Accords and SALT II. But it was a pretty serious blow to it, in both Washington with the rise of human rights in foreign policy in general, with those who wanted to alter Soviet internal rather than external policy (the Russians did not like Carter's focus on that, in spite of the fact that he was less "hawkish"), and the Reaganite right in the GOP, and in Moscow in giving credence to people who wanted to pursue a more aggressive policy in the Third World and thought the USA was irrevocably weakened.
Sources:
Anatoly Dobrynin-In Confidence.
Vladislav Zubok-A Failed Empire: The Soviet Union in the Cold War
Steven Ambrose: Nixon, Volume III
Jussi Hanhimaki: The Flawed Architect (Kissinger)
The Mitrokhin Archives.