I noticed one of my logistics fleet is going to my second star system, and has a load with a "skull" near to the label. Please, what does this symbol mean and what are logistics fleets used for? 

Thats the population it is transporting to another system.

Do logistics fleets only transport populations including minor factions?

Pretty much, you might occassionally see one transporting food to outposts, but as a Vaulter I have no such problems.

Logistic ships transport food or population. This one particular transports 1 Haroshem pop (it's their icon), food whould have its own icon.

Typically when exploring curioisities, you discover minor faction population. They then travel to the nearest system with open population slots using Logistic ships. Logistic ships transport Food or Population. What you have there is exactly that and is also what you would see if you were manually moving population from one system to another via the Spaceport(which you get with a level 2 System or higher).

You need the required research level in the right quadrant of the research wheel, then go to the economics screen (where the market is), select the respective system development upgrade, and choose which resources you want to spend on that (you'll have to pay per system). I recommend the wiki page. 

GTI wrote:

Barf wrote:

(...) if you were manually moving population from one system to another via the Spaceport(which you get with a level 2 System or higher).

How do I raise the level of a system?

Unlock Tier 3 of Economy and Trade quadrant in Research Tree(East quadrant or right side). Doing so unlocks level 2 System Development which allows you to spend Luxury Resources to apply a specific buff to your systems. What you choose can not be changed so choose wisely and only the Vaulters can use Strategic Resources in lieu of Luxury Resources in System Development. To choose your bonus, access the Economy page(Gold icon at top left of page) and youll see it right away. Popular choice for level 2 is Jadonyx since it applies a +60 Industry bonus to the system. Then when youve chosen your development, it will appear in the list of build options in each system. Queue it and the system will upgrade once complete. Level 2 takes 1 turn, level 3 takes 2 and level 4 takes 3 turns to finish. To unlock level 3 and 4, simply unlock tier 4 and 5 of Economy and Trade respectively.

Thank you for all the replies people!