And in particular, why off world? There is a lot of garden world territory available inside the Imperium and forming a colony on an unsettled part of a moderately populated garden world seems like a much better choice than attempting to set up in a new system. Possibly, I suppose, because the government (or governments) on that planet have already claimed jurisdiction over the entire world, populated or not. 

There are 273 garden worlds in the Imperium with population densities of less than 1 person per square km of land area, and 22 with populations of less than 10,000. 

Much like tech startups in the bay area, I do imagine that much of the frontier areas of the Imperium are littered with forgotten attempts at kickstarting a colony by an under-resourced group if the main purpose is to set up their own government because they don't like some aspect of either the Imperial government or their world's government, but that would have to be pretty extreme. Pretty much every election cycle in the US results in one side or the other declaring that they're moving to Canada - which is a far easier prospect than moving to Ceti Alpha VI - yet the count of people to have actually done so seems to be extremely small. 

I do like the idea that many Imperial worlds are prone to spurious religious groups forming (Church of the Seven Stars) and this does often provide a driver to colony attempts, and if we accept that gems or other planet-only resources are in demand for whatever reason (though I can't imagine that a TL15 civilization couldn't produce them en masse from raw materials in city-sized labs if they wanted) then prospecting bases that turn into small colonies would spring up now and then. 

In terms of social development, I wonder how frequently the common knowledge of typical things about the Imperium gets lost. A small colony sets up (perhaps without informing the imperial government, or outside the domain of the Imperium) via contracted transport of the colonists to a world and after a few curious free trader visits there are no more. The colony is more or less on its own. Children born to that colony will probably be taught that there is a huge empire out there among the stars that they came from, but none of them are likely to ever visit it and they won't be able to relate that first hand experience to their own children. Teaching might shift to more practical matters ("make sure to get the harvest in before aphelion") and most of the concept of the Imperium gets lost completely. There's not really that connection to stabilize social development and I imagine that the colony will rapidly (a couple of generations in) develop its own culture and view of the universe.