Expansion, highly dependent on Exploration in Starspan, boils down to the notion of making your spacefaring empire bigger through colonization or conquest.
There are benefits and costs to expansion; a bigger empire has more resources, but managing more people, resources and colonies becomes more and more difficult. As empires expand, problems of corruption grow and they run the risk of upstart factions taking control over parts if control is not carefully maintained.
The first and seemingly easiest method of expansion is colonization. Colonization can take the form of creating a new colony on an already habitable planet, establishing a sheltered colony on a hostile planet or minor planet, or building a new habitable space station.
There are several types of colonies, each with a different purpose. Each has some requirements to plan and launch that, at the beginning of the game, can be thought of being on par with a great project. Civil engineers and scientists are needed to determine what the target site's needs will be to establish the colony ... will the soil support seed to grow food? How seismically active is the environment if it is on ground (a planet or minor planet) -- will buildings need to be extra tough? If its in an inhospitable environment, what will the domes have to contend with ... radiation? heat? All these elements need to be analyzed by scientists who conduct survey missions and planned for by the civil engineers. What specialists will the colony need? Government officials? Construction workers to help expand? Agricultural experts? They don't just pop out of the population magically, you need to produce these types of specialists through education and training from the world you will launch the colony expedition from.
You may find, at times, there are no good places to expand, and a rival civilization either has a colony you need to meet the needs of your empire, or they are weak (or have angered you sufficiently) that you simply wish to deny them that resource. As some politicians of late have discovered, Conquest is not a simple matter of smiting the enemy military; you will need a way of controlling the conquered population by either assimilating them into your nation's culture or finding a way to ... well ... dispose of them. Do the inhabitants of the world you plan to conquer even breathe the same air or eat the same food? You can have more than one species in your empire, but each needs their own accomodations depending on their life support requirements. You need people to take over and manage the conquered world, and it would be useful to have engineers who specialize in reverse engineering alien technology to try and decipher all the alien technology you can plunder from the world. Again, this does not magically happen unlike other games.