Shortage of rubber

@user6760? Okay?

A basic biological imperative is that humans have more children when general mortality is higher. This is entirely instinctive. I'm assuming the Spacer's environment is more dangerous? You don't need a different reason.

@CelestialDragonEmperor Because you seem confused, he means condoms, the most common birth control.

@Anoplexian oh! Now I feel stupid because that joke went over my head

"I don't think simply wanting to have more than 3 kids without buying a permit is a good enough reason or motivator to move to space." There are definitely people who would think it is.

@eyeballfrog yes but I doubt they would uproot their life just to not pay a small fee for having a third kid.

@CelestialDragonEmperor Clearly the permits are expensive enough to be a burden, otherwise they wouldn't be a very effective deterrent.

Not necessarily the most serious answer, but maybe low gravity sex is just more fun.

A genuine explanation could simply be wealth. In real life poorer countries tend to have higher birth rates, rich families tend to have fewer kids than their poor neighbors. If you send all the plebs to the space colonies to mine unobtainium while you let the rich folks enjoy the last bits of habitable Earth there's a good overlap in the space-population/poor-people Venn diagram, explaining perfectly why birth rates in space are higher.

minor nitpick on "usual on earth (1 or 2)". if with usual you mean average this is wrong. It is above 2, otherwise our population wouldn't grow

@Ivo the average for the Hegemony is 1 or 2 with some people having more via birth credits

Boredom during the long flights. What else would you do in zero gravity? Also gives your story a whole new angle ;-).

Make the spacers more religious than average. Even if an individual character is not especially religious, religious societies tend to have higher birthrates. You can throw in some jazz about contemplating the infinite in space making people more pious, too.

Abstinence-only sex education in spaaaaaaaaace

Assuming space-jobs pay more on average than Earth-jobs, it's more likely they can afford a third-kid permit. If the permits effectively lock out most of the Earthbound population due to price alone, you don't need any other explanation.