Harvesters convert the sorium straight into fuel which fills it's own fuel tanks.
There's a few ways to transfer fuel, the most immediate way is to open up the individual ship window and select either ship you want to transfer from/to. Then its similar to ordnance transfer, click on the miscellaneous tab, you'll see "Manual fuel transfer (ships)" and "manual fuel transfer (population)". Ship's has a menu where you can select any ship at the same location, enter a number under "transfer amount" and hit "transfer to", or "transfer from". Population is the same but without a selection, since you can only orbit one planet at a time :p.
Now the other method of transferring is with task group orders, a ship designated as a tanker under the class design window has additional order options. Any ship can "refuel from colony" at any colony or "refuel from target vessel" at a tanker, and a tanker has additional order options for "unload 90% fuel to colony", "load unload fuel to set level" and "refuel target fleet".
So your fuel harvesters should always be designated as tankers so that other ships can easily refuel from them.