Why was the US the only country to go the moon.

because the powerful russian rockets that were supposed to take them kept exploding

If we won the race, why would it make sense for them to keep running?

There are many reasons, but the two main ones were

- money

- priority/leadership

And they boil down to one decision and one event.

In the USA, following president's Kennedy's decision, all the eggs where thrown into one basket: NASA. All the space program was coordinated under one agency.

In the Soviet Union, there were three programs, each one under a different agency.

That was the decision.

The Lunar program was under Korolyov, a brilliant rocket designer. The Soviet Lunar program was way ahead until... Korolyov died.

That was the event.

This fact was not known at the time (in fact, it was still secret, when it was finally revealed decades later). After his death, the Lunar program continued on, but no one could get the heavy launcher to work properly.

Meanwhile, the unmanned exploration programs continued (under other directors) and the Soviet Union continued to be first on many other fronts (first soft landing on Venus for example).

But the manned lunar program never recovered and, after the Americans made the second manned landing on the Moon, the other program directors convinced the authorities to stop spending on the manned lunar program, and turn back to what the Soviets were good at: unmanned probes to other planets.

They still ran a few (unmanned) lunar programs, including the return of samples from the Moon's surface (3 missions) including one automated rover.

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First country in space: Soviet Union (Sputnik 1, by Korolyov)
First man in space: Soviet Union (Yuri Gagarin was the astronaut)
First Woman in space: Soviet Union (Korolyov was directing)
First photograph of the other side of the Moon: Soviet Union (Korolyov's project)