You're on the right track. The short answer is: insulin moves energy into fat; lack of insulin allows energy to move out of fat.
In a healthy individual, this isn't a problem, like kenckar says, between meals, your insulin will come down and you'll burn that fat you stored. What Taubes is most concerned with is in the metabolically deranged / insulin resistant person. In these people their insulin is chronically elevated so basically they never get that energy back out of their fat cells and just keep putting on weight. That's also why someone who's insulin resistant can be ravenously hungry and feel like they're starving even if they have the fat; they can't use it, so their brain is panicking and telling them to eat more.
This is why Taubes has come to the conclusion that insulin is the one hormone to rule them all (for fat loss). If you can control the insulin, you can control your available energy. That's a hot topic on PH, so I'm not going to argue either way.
On to your example, If you need 3500 cals to be as active as you are and you ate 1750 as low carb: you're right, you'd just pull the extra out of your fat for the day, losing about 1/2 a pound.
But if you ate 1750 of carbs, some of gets used for fuel right away, the other part gets shunted to your fat cells because your insulin goes up. What happens next is the question. If you're a healthy individual, your insulin will come back down and you'll pull what you need out of fat and essentially lose another 1/2 pound or so. If you're insulin resistant, then you're out of luck. You can't pull that energy out, so now you get crazy hungry. If you don't ever get around to eating, then your body has no other choice but to slow its metabolism because there's no energy around to use. This is where Taubes says "you're not fat because you're lazy, you're lazy because you're fat". I.e., you just can't move because the energy isn't available for you to use.
You'll also notice something in that example above with the insulin resistant person:
- They need 3500 calories
- They eat 1750 calories
- They use (say) 875 calories immediately from blood sugar; the other 875 gets stored because insulin was/went up
That means, they took in what they thought was 1/2 of their caloric need, hoping to lose 1/2 of a pound, but they ended up storing 1/4 of a pound worth of energy and only getting 1/4 of their need for the day actually used. That's the kind of thing you see when insulin resistant people try to diet (by cutting calories but keeping the carbs up). In this case, they effectively only got 1/4 of their daily need for energy from their food, they gained some weight, and they had to slow (sometimes irreversibly) their metabolism down. I think that's why the food quality matters more than food quantity when losing weight (of course, you do have to know the difference between your mouth and vacuum cleaner).