Actually these resources are on a planet of that star system.
You see, there are two types of resources in this mod. Those commonly found in asteroid fields like Hydrogen, Titanium and Crystals, and those on planets like Cattle, Fish or Cotton. The former are for the most part too far away to be reached by city culture in a realistic timeframe, which is one of the main reasons Starbases exist. They can be constructed by a Construction Ship which gets consumed in the process for a sum of money and are basically the strongest units in the game, with only the mighty Battleship standing even a chance of standing up to them. However they are immobile, being space stations and all, but they provide culture or influence as it is called here in a two tile radius, with the third and final level three tiles. Note that their radius also includes the four tiles in the corner unlike regular city culture. Build them near a group of resources you want, preferably being in an asteroid field itself for the extra defense, and you can build Extraction Facilities on the resources in their radius to gain access to them once they are connected to your empire via warp or jump lanes. Be aware that the cultural value they give to the tiles in their radius is 0%, meaning that as soon as the culture/influence of an actual city reaches them they instantly flip to them.
The planetary resources are all on, well, planets. To benefit from them you need to colonize their system, i.e. build a city, and connect that to the rest of your empire. All planetary resources also have an affiliated building that can only be built on their planet, e.g. Ranch for Cattle, Winery for Wine etc., which are all pretty useful. I'm not sure if they are actually needed to connect the resource, but I build them anyway as one of the first things in a new city if the resource is on one of the first three planets.
Planetary resources all give only a small health or happiness bonus, while the ones in asteroid fields mostly boost production speed of certain unit types, except for Oxygen (2 or 3 health), Water (2 or 3 health), Crystals (2 or 3 happiness) and Uranium (or maybe it's called plutonium, not sure, enables build of this mod's nuke equivalent).
I forgot to check which color my star has this turn. :/