Great question. I highly recommend you read Radchenko's Two Suns in the Heavens: The Sino-Soviet Struggle for Supremacy, 1962-1967. It's incredibly accessible and very comprehensive. I'll try to answer your question succinctly below with references to that book and some other great sources that can be found at the Wilson Center's Digital Archive (link). I've broken my short response into two sections. The first briefly goes over one of the main roots of the eventual split. The second then explains why China and U.S. eventually normalized relations.

The Sino-Soviet split is fascinating for several reasons, including the seemingly contradictory nature of it all. In 1950, the two countries signed the Treaty of Friendship, Alliance, and Mutual Assistance Between the People’s Republic of China and the Soviet Union. By the late 1960s, however, the two countries had massed forces along their gigantic shared border. These tensions eventually boiled over into direct military conflict in 1969. Not only joined ideologically against the imperialist world system, China and the Soviet Union actively exchanged goods, arms, and technical specialists to further strengthen the international communist movement. How then did these two countries reach such a hostile point?

Many of the issues which ignited the Sino-Soviet split are rooted in Khrushchev’s secret speech following the death of Stalin. Delivered on February 25th, 1956, Khrushchev’s speech outlines several examples of Stalin’s negative impact on Soviet development and the Communist Party of the Soviet Union (CPSU). While the PRC initially responded warmly to the speech, what soon emerged was a contest for the leadership of the so-called "global communist movement." On one side, you had an increasingly radical and paranoid Mao who believed the USSR under Khrushchev had turned away from global revolution thereby disqualifying itself as the leading communist country. On the other side, you had Khrushchev who was seeking to purge Stalinist elements from the party-state apparatus while repairing relationships between the USSR and other communist countries (namely Yugoslavia, Albania, and China) and also maintaining the primacy of the USSR in the global communist movement.

This conflict can be clearly seen in a pair of open letters that were published in June and July of 1963. In June of 1963, the CCP published an open letter laying out the case for why and how the CPSU had abandoned the revolutionary cause and instead committed itself to revisionism and great power chauvinism. The letter continues by describing how the CCP, with Mao at its core, are the sole true revolutionary party capable of leading the global communist movement. Already isolated from the West, China's audience with this was other communist countries and the wider developing world. The following month, the CPSU fired back with a behemoth of a letter. In it, they defend the USSR's leadership of the global communist movement and list the extensive aid they have provided China. They then go on to detail how "there came to light serious differences between the CPC on the one hand and the CPSU and the other fraternal parties, on the other.” Reasons cited include China's increasingly erratic behavior, its incessant polemics against the USSR, and the CCP's fundamental misinterpretation of Marxism-Leninism. Relations between the two countries and the two parties only deteriorated further until open military conflict broke out in 1969 (as I stated above.)

So, where does the United States come in to play? Simply put, the enemy of my enemy is my friend. As relations between the USSR and PRC plummeted, Mao and the CCP soon realized that their military and economic dependence on the USSR and their diplomatic isolation from the developed world was a serious problem. To alleviate this issue, they began looking to the West. An April 1969 cable sent by the French ambassador in China to his counterpart in London perfectly summarizes the PRC's thinking. Amb. de Courcel details how China is increasingly interested in establishing diplomatic (and economic) ties with "Canada, Italy, and maybe other countries, which would strengthen [China's] position for eventual discussions with the United States." Not long after this cable was sent, Kissinger began his secret back channeling with the PRC culminating in Nixon's famous visit in 1972. From the United States' perspective, establishing formal diplomatic relations with the PRC achieved two main goals. First, it would help bind the erratic PRC to the liberal international system in the hopes of deescalating any military conflicts and eventually introducing some type of democratic and/or capitalist reform within the PRC (although this second point is still hotly debated. See Jerome Cohen's recent essay for a fantastic overview of the topic.) The second goal was the weaken the position of the USSR. If the United States could welcome the PRC into the liberal international order in some form or another, it could further exacerbate frictions between the PRC and USSR thereby diminishing the threat from both countries, but especially from the USSR. By 1978, the U.S. and the PRC had normalized relations. The following year, Deng Xiaoping visited the United States and the United States granted the PRC temporary most favored nation tariff status. And the rest is history!

Was the CCP contradicting itself by going through with normalization? Of course! Did normalizing relations with the U.S. and the rest of the West work out in the PRC's favor? Of course! To use a tired Deng Xiaoping quote, "It doesn't matter whether a cat is black or white, as long as it catches mice."

I hope this answers your question. This is my first time responding to a post so I apologize for any and all subreddit rules I am breaking.