The blue-white circle icon shows spots where you can refuel. Typically your colonies (with fuel in store), and gas mines. However, there may be only a limited amount of fuel, and if your fleet is giant, there is not necessarily enough fuel for all. I think fleets in general avoids refueling at a station/base where there is not enough fuel to refuel all.
For large fleets, in addition to enough fuel, it help to have more docking bays so that more ships can refuel at once. Gas extractors are for getting the fuel from the source (gas planet) and into store, refueling happens from the cargo bay of the station/colony. The only thing docking bays helps with is refueling faster, though, not finding fuel in the first place.
For better fuel capacity (or at least range) you have: More fuel cells. More effective fuel cells (research), more efficient reactors. And probably more efficient engines, hyper drives, and weapons too.
A troop transport sitting outside a colony is probably doing just that. Sitting there. If it has no order it does nothing if not automated, and even if automated I think troop transports are prone to sitting around once full with troops.
I can't say what is the problem in your game, but I can hazard a guess that you either have way too large fleets, too few fuel sources (gas mines on fuel sources), or a mix of both. In war, the AI attacks mines which it seems as more valuable, and which is knows of. Gas mines are typically among the more valuable... Maybe your gas mines have been destroyed by your enemies, and no new ones are built because your constructors died too?
How do I manage fuel?
I refuel by the icon below the selection panel, or by right click "repair and refuel". If they pick a destination far away, I may click on my selected fuel source and choose "refuel" so that they head there, even if fuel is a bit short. If only a few ships are low, it is also possible to give refuel order to just those ships (but make sure you don't interrupt their refuel order by an fleet order before they have refueled).
I usually leave my constructors on auto and let them do their stuff. In addition I have a habit of always manually booking a fuel mine for each new colony. If not in the colony system, so in one nearby. I book the mines by locating the source and selecting it, choosing the button in below the selection panel that says "queue nearby constructor to build gas mine her" or something to that effect.
To prepare for war, I usually build and deploy a resupply ship or three in relevant areas. Relevant areas are in the staging grounds of the fleet (typically near the edge of the empire, not in the fuel rich central parts), and in quiet corners nearby to my targets for the war.