How the space race ended?

Answer 1

The space race essentially ended once America had landed on the Moon. The USSR claimed they'd never actually wanted to go to the Moon (although it was later confirmed that they certainly did have a Moon mission planned, but the rocket that would have been used was never considered safe enough for manned use).

Once the Moon had been reached, there were two conclusions the US public reached:

1. JFK promised the USA would reach the Moon before the end of the 1960s and, by implication, before the USSR. That was achieved. So, the USA "won" the race - even though the USSR tried to claim there hadn't been a race.
2. The Moon landings were incredibly expensive and, apart from the prestige, without enormous benefit.

These two facts were enough to bring about the cancellation of the final 3 Apollo flights.

Answer 2

The space race ended with the culmination of the Apollo program in 1972 when Apollo 17 became the last manned mission to land on the moon. This marked the United States' victory in the space race over the Soviet Union. After this, both countries shifted their focus towards cooperation in space exploration rather than competition.