Why do they want to hire prostitutes for the space program?

It's actually from the novel:

The film adaptation follows the source material fairly closely, with a couple of exceptions. The view on women has been updated. In Heinlein’s story women’s role in space travel is limited to providing sex and company to male astronauts, basically as some kind of luxury prostitutes. In the movie they’re more capable than that.

Drew McWeeny from HitFix discusses the subject with Ethan Hawke (around minute 5) and they agree that the movie tried to keep close to the source material, i.e. the idea of space prostitutes was a very Sixties idea.

It's a real life thing, sex slaves, I mean ""comfort women"" that are found in any large military/army camp to keep the men happy/in-line. In modern times, it's less heard of, but still common.

Japanese Comfort Women:

[...] were women and girls who were forced into sexual slavery by the Imperial Japanese Army in occupied territories before and during World War II.

The name "comfort women" is a translation of the Japanese ianfu (???), a euphemism for "prostitute(s)". Estimates vary as to how many women were involved, with numbers ranging from as low as 20,000 (by Japanese historian Ikuhiko Hata) to as high as 360,000 to 410,000 (by a Chinese scholar); the exact numbers are still being researched and debated. Many of the women were from occupied countries, including Korea, China, and the Philippines, although women from Burma, Thailand, Vietnam, Malaysia, Taiwan (then a Japanese dependency), Indonesia (then the Dutch East Indies), East Timor (then Portuguese Timor), and other Japanese-occupied territories were used for military "comfort stations".

South Korean government who made women available for stationed allied US Military:

A group of women in South Korea sued their own government in June, alleging that it trained them to serve as “patriots” or “civilian diplomats” in the 1960s and 1970s. Their real job: work as prostitutes near American military bases. The women were tested regularly to make sure they didn’t have sexually-transmitted diseases, and were locked up until they were healthy again if they did, they said.

It’s an uncomfortable part of the U.S. military’s long history with prostitution. The world’s oldest profession has long catered to U.S. troops, whether at home or abroad. But the issue is getting new scrutiny in South Korea, where the top U.S. commander, Gen. Curtis M. Scaparrotti, recently forbid all military personnel under his command from paying an employee in an “establishment” for his or her time.

The Nazis too:

German military brothels were set up by the Third Reich during World War II throughout much of occupied Europe for the use of Wehrmacht and SS soldiers. These brothels were generally new creations but in the West they were sometimes set up using existing brothels as well as many other buildings. Until 1942, there were around 500 military brothels of this kind in Nazi occupied Europe. Often operating in confiscated hotels and guarded by the Wehrmacht, these facilities used to serve travelling soldiers and those withdrawn from the front.

And that's just the last 60 years? There is a long history of sexualized violence, which even "consensual" prostitution is considered. This is borne from the idea that women are things for men's enjoyment or comfort and not people, still very much prevalent at the time the novel was written, 1958.

It's a very common trope seen in a lot of science fiction, based on the idea that the people most likely to be willing to give up their current lives to explore the new frontier of space (which could require a commitment of years and years, possibly life, depending on the end point) would be young, adventurous single people, and back when a lot of these tropes arose it was just assumed they would mostly be men. But what is the one thing many young, hardy men would never give up? Sex. So they add prostitutes (also usually single and thus often willing to attempt this new life if the price is right) to the equation to make sure men will sign up for the program.

Remember this isn't like say... exploring the West Indies in the Age of Discovery, where sailors could assume they would run into inhabited places eventually and there would be access to sex. This is space.