How was Reagan’s Challenger speech influenced by the Cold War?

The Cold War and its tensions are subtly woven into the fabric of Reagan's Challenger speech. Primarily, the speech is concerned with mourning the astronauts tragically killed in the space shuttle disaster. Reagan also wanted to reassure a shocked nation that though this was a setback to the United States' space program, it certainly wouldn't be the end of the matter. American space exploration would continue.
Reagan's confidence in the future of the space program wasn't just an expression of facile optimism; it was a clear message that the United States would continue to compete with the USSR in the space race—an intensely fierce competition that the United States didn't always win.
President Reagan never mentions the Cold War explicitly in his speech, but the oblique reference to US–Soviet tensions in the following remarks is inescapable:

We don't hide our space program. We don't keep secrets and cover things up. We do it all up front and in public. That's the way freedom is, and we wouldn't change it for a minute.

At the time of the Challenger disaster, the Soviet Union's rival space program was still highly classified. Reagan draws attention to this fact, though without directly mentioning the Soviets. The United States space program, with its openness and accessibility, is used by Reagan as a symbol of American freedom, a freedom he implies is not shared by the Soviets.
https://history.nasa.gov/reagan12886.html