Let me see if I can help. (And PLEASE IF I GET ANYTHING OBVIOUSLY WRONG, SOMEBODY CORRECT ME!!!!)
At the top of the screen is a set of 'information blocks' which tell you how much money, manpower, war supplies, general supplies, and ammunition you get each turn. Took me a while to figure out, but if you rest your cursor over a region,
another set of info blocks appears below these which tell you how much of each of these this particular region supplies to those totals.
So, "supply" is generally the 'general supply' category, which I envision as 'food', uniforms, boots, buckles, buttons, and so on, plus ammunition. Now any given region produces only a tiny fraction of the total. (As an example, the CSA generally produces about 30 conscript points per turn and these are produced in only certain 'structures' which may be identified by having little white 'tents' in the 'town square'. These I envision as recruitment centers. Only Richmond produces more than two, although that CAN be altered by the placement of an officer with the 'recruiting officer' trait. And the rest come from diverse places scattered all across the map, one per state, as near as I can determine. Similarly, War Supplies are produced in only certain regions, in this case, always with structures. General Supplies and Ammunition are more spread out, but again are produced in specific regions. Thus, say, Winchester, VA produces perhaps 16 general supply and 8 ammunition points per turn. NOT enough to supply a major army for any length of time. Manassas on the other hand, while it produces the same amount (and these amounts may be different, I'm just offering a scenario) and shows this in the lower set of info blocks at the top of the screen, it ALSO says that the available stocks are much higher say, 67 and 40, if you rest the cursor over the structure itself. This is because it has a depot, which are generally found along major transport lines, such as railways or navigable rivers. (OR wherever you happen to decide to build one, such as, say, at a 'chokepoint' for supply. Which would be a region along a major transportation line, such as a railroad, which has no level 3 or above structure or depot and therefore cannot forward supply during the 'supply phases'. Also known as 'the place where supplies go to die'.

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Where does all that extra supply come from? All that 'general supply' produced in places like Savannah, GA, or Birmingham, AL gets collected, if surplus to the regional needs, it is 'sent forward' to where it's needed. Locally, this might be anywhere from 3 to 5 regions, depending on weather, the type of transportation available to move it, and the percentage of control of the region. For areas farther away, transport is ALWAYS by rail or riverine transport. As I understand the method, the number of transport of a particular type, say rail, is taken, the number of units of that type that are dedicated to moving troops is subtracted, and the remainder is available to transport supply. (I'm unsure of the mathematical algorithm used, as in 'how much supply per unit of transport, but hopefully you get the idea.)
This supply transport is handled in 'phases', (these show up during the turn resolution as 'Supply Phase 1' through 'Supply Phase 5' up at the top of the screen.) Now I'm treading on a bit thinner ice, but if I understand it, in phase one, it moves supply forward up to 5 regions, if there is available transport AND there is a level 3 or above structure OR a depot within that range. If there is no level 3 or above structure or depot, then no supply can transit that route. It will test all the available routes and move supply forward on those that meet the test. This repeats from the new locus (the point to which supply was moved in the previous phase) to another level 3 structure or depot within a maximum of 5 regions in each succeeding phase until either the maximum number of phases have occurred or the front lines have been reached.
The 'maximum 5 region' rule applies to rail or riverine transport, if I understand it correctly. If using road transport, I believe the maximum is 3 regions; if no road, rail or navigable river exist then I believe the maximum is one region.
Supply wagons represent the army 'train' which was a bunch of physical wagons (hence the representation, how historically accurate!

) These got loaded up with food, clothes, medicine, boots, buckles, hats, coats, bibles, sergeants chevrons, officers braid, and what not, and trailed along behind the army to resupply them as they marched. IF the army with which they are traveling is within 'supply range' of a level three structure or a depot, (from three to five regions over which the side that army is associated with exerts some minimum level of control and depending on both weather and the type of transportation available), then the wagons remain 'full'. (I envision this as wagons shuttling back and forth to some supply head, a rail depot, river pier, or a town.) If not, the contents of the wagons begin to be drawn down, represented in the game by the number of boxes and barrels shown on the unit being reduced from two each to one each to none. If there are sufficient supply wagons available, then in a situation such as a siege within enemy territory the individual supply wagons may be shuttled back and forth to the nearest supply head in a 'rotation' which provides the army with it's supply. (I believe that this represents some of the 'micromanagement' that generates so much comment on here.

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I believe this is a reasonably accurate description, although there are little 'fillips' such as that the Union player may use transport ships in place of a 'supply head'. (An army relying on this still needs to have supply wagons with it, but the ships apparently may take the place of a 'level three structure or depot'). (And the transport ships have to be shuttled back and forth to some port to 'refill', unless the player puts them in the Atlantic Shipping Box, in which case, they are apparently assumed to be shuttling supply to the army in question.) This is very useful for amphibious invasions.
The same sort of thing is theoretically possible with river transports (the physical ones represented by ships that are purchased as units, not the 'riverine transport' represented in the transportation assets,) but I've not seen it used by the AI and have never had occasion to try to figure out how the mechanics of it worked on my side.
I've probably got half this wrong, but for what it's worth, here it is. Hope it helps.