Speculation alert, as usual on this topic.
Part of the concept involves the collection of nitrogen, which is a vital element for life support in space. The atmosphere is nearly 80% nitrogen, and it is there, so you may as well use it. The trick is to get around the energy problem. In order to collect the gas, you need to compress it and cool it down considerably. This takes a lot of energy.
Rather than use a lot of energy to cool down gases, perhaps it would be better to minimize that. By using a hydrogen fuel cell, you can still collect oxygen from the atmosphere, and then use it to make electricity and water. That means you have to crack it later, but you still have the oxygen in a form that is easily stored while making energy- as opposed to consuming it.
Can you not apply that principle to using the nitrogen, which is also present in the atmosphere? If you can't use it, you have to dump it, which is a waste. Why not find a way to make this useful as a propellant as well while not having to put it into a form which consumes so much energy?
The nuclear powered version of this concept uses the nitrogen, but nuclear power in space is a problem.
Making energy from the oxygen in the atmosphere is definitely not a new concept. Airplanes do it. The only question here is if there's enough oxygen that can be collected -so as to generate enough thrust for overcoming drag- which is the same principle that applies to airplanes.
Is nitrogen worthwhile as a propellant? Yes it is. But how to make nitrogen based propellants in LEO? Let look at the proposition, shall we? Let see if there's a way to use nitrogen- by collecting the nitrogen and the oxygen in LEO, and then combining them somehow for power, or for thrust.
If a nitrogen fuel cell were possible, you could make the electricity as a hydrogen fuel cell does, and keep the oxidized nitrogen on board for processing later. If it could be converted to something more easily stored to liquid form than pure nitrogen, that would be helpful with the energy problem. (It would be easier to store it as a liquid than as a gas.) Nitrogen fuel cells may not be possible, so far as I know.
But there are a couple processes that could lead you to a form of nitrogen that could work, at least theoretically. The hard part may be in making these processes fit on-board a spacecraft. Provided that you could do that, then store the nitrogen via nitric acid solution in water. The Ostwald and Haber processes are what I had in mind.
The biggest trick of all is to do all of this in real time as the spacecraft is skimming the gases at the edge of the atmosphere. You have to be able to get more gases into storage than what you are using them for in making energy, while at the same time not having to dump much matter overboard. That means the conversion to nitrates are sufficiently exothermic to help support the conversion.
The hydrogen fuel cell accomplishes this by producing electricity as well as water. The two processes mentioned in the previous paragraph will utilize the nitrogen from the atmosphere in order to make the water into an nitric acid solution. Those processes are exothermic, but are they sufficiently so? Another problem is the hardware- is it too complex and is it too massive?
Assuming those questions can be answered, you're in business. Once the tank is full, the acid can be offloaded in higher orbit, where it can be processed further into rocket fuels.
The entire system could do this over and over again so as to create a steady stream of product. It would need to have two components launched from the ground-- the machines that do the processing and the hydrogen. The machines could be launched once. The hydrogen will have to be launched at a rate sufficient to support the entire operation. Given that hydrogen is the lightest element, it will be advantageous in offsetting the mass penalty in spaceflight.
The mass penalty is a major stumbling block in making space more accessible. Given the mass savings, the cost of getting fuel into orbit could be reduced by perhaps as much as 1 order of magnitude from the oxygen alone.
Update:
Earlier versions of the LOXLEO thread included the use of a VASIMR as the propulsion engine. If additional energy is needed for the above mentioned chemical reactions, perhaps the thermal management issue of the VASIMR can be turned into a resource. A heat exchanger can move the heat from the VASIMR- where it may be a problem- to the reaction chambers, where it may come in handy.
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