Starting from simplest to most complicated, lets tackle the size issue first.
Large insects physiology
Insects do not breathe with lungs the way we do. They have many spiracles distributed over their body that open into many tracheae which allow oxygen into their body. They can't use expanding lungs the way we do because their exoskeleton cannot deal with that much flex. Studies have shown that the ability for large insects to evolve is highly limited by their ability to get oxygen to their leg muscles. Increase oxygen, such as in our Paelozoic era, and you increase their ability to get large. You want a Paeleozoic era level of oxyegen in the atmosphere of your planet.
Low gravity would also help large insects be realizable. While it appears large insects can evolve in Earth like gravities, they are bound by the Square-cube law, and it seems more likely that exoskeletons will remain a valid solution as they get larger.
An unfriendly planet will increase the viability of exoskeletons. The big advantage of exoskeletons is that a strong strike from the world around you does no damage as long as it wasn't sufficient to crack your shell. Endoskeletons will be more desirable in situations where it is valid to take a little damage from a strike, and then heal it afterwards. If the environment is hostile, with many potentially killing blows around every corner, shielding will be more useful than repair.
Culture of a Hive Insect Population
The defining characteristic of the hive mind is an unusually strong correlation between the actions of individuals, apparently centered around the mind (historically depicted as the "queen" of the hive). In order to do this, there must be communication -- constant communication. Much of this communication is built at the genetic level (especially given the short lifespan of bees, giving few opportunities for learning). This is where your question gets interesting. Going to the moon is a HUGE communications gap, which limits this communication. This is going to drastically decrease the culture's desire to travel to the moon. It's just not the nature of a hive mind's mentality.
Two directions which could help are RF technology and genetics. Consider a world colonized by one or more of these hives (you did not specify if the entire species was a hive, or if there were multiple competing hives, like ants). They would eventually reach a balance, and live in the best harmony the planet can afford. The discovery of RF communication could dramatically increase the rate of their communication, and allow it to occur through the vacuum of space. This would, in a flash provide a new surface to colonize and trigger a space race.
However, a hive mind would use communication as simple as ours, so it wouldn't be as simple as a phone conversation. Hive minds require coherency, and the easiest way to do this is analog communication such as sounds or dances (digital brings up all sorts of timing woes that, in my opinion, prevent a hive from functioning adequately). Just because they had a new technology like RF wouldn't cause them to immediately shoot for the moon. They would need to adapt the technology to their communications.
Thinking about how such a race would become so powerful, their individuals would have to be well programmed. Such a race would probably need a way to genetically wire their workers to listen to the right commands and act accordingly. They would probably conquer genetics and biology long before RF. Accordingly, they could design workers that would be custom engineered to respond to RF communication just like their primitive songs and dances.
I would also expect them to split the difference between biology and technology: I would expect the technology to be physically adapted to be more similar to their primitive songs and dances. How much of this balance is genetic, and how much its technological is pretty much up to you.
I would consider looking at more complicated hive mind structures than the simple queen+workers pattern, because there's a lot of room for creativity there. If there is a multi-level hierarchy, the queen might install a lieutenant on the moon, equipped with powerful RF communications. Alternatively, a simple queen+workers hierarchy might simply be looking for a backup plan in case something went wrong on their planet. Being able to evacuate the queen to a moon already colonized by workers could be a strong strategic move.