A few things:
There are three tissue types reactive to insulin, in order of sensitivity from most to least: Liver, adipose (fat), and skeletal muscle. While other tissues are weakly reactive to insulin, they don't need it to take in glucose for metabolism. Nervous tissue requires no insulin at all to absorb and metabolize glucose.
The glucose regulation system is rather complex. The "lay" explanation given to the public is vastly oversimplified, so most people believe that insulin is pretty much the only thing operating in this regulation, and that all cells need insulin as a "key" to take up and the burn glucose. This is grossly incorrect, and it is this simplified view that makes things like what you experienced seem so paradoxical.
Diabetics are generally a bit better educated about glucose regulation, especially type 1's. However, one aspect of glucose regulation that is not well understood even by most diabetics is hormone signaling during digestion and fasting, and the impact it has on blood glucose.
During normal fasting periods between meals (3+ hours after eating), when you're not hungy, the liver simply breaks down glycogen via glycogenolosis, releasing glucose into the bloodstream at whatever level is necessary to maintain steady BG as the body uses the glucose.
However, when you go too long without eating, signaled by feeling hungry, the brain and gut signal the liver to dump more and more sugar into the blood. When you take insulin in excess of basal amounts, that stimulates sugar uptake primarily by the liver which assembles it back into glycogen complexes via gylcogenesis.
This is going on at the same time the liver's breaking this glycogen down and dumping it into the bloodstream. Meanwhile, the liver is also breaking down the insulin and removing it from the blood.
So, when you starve yourself, get rising BG, then treat only with insulin, you are in essence having your glucose metabolism "run in circles". The same glucose cycles in and out of the liver round-robin, consuming the insulin, but on net no glucose is being removed from the blood.
This will go on until you eat. When you do, the gut releases other signaling hormones that tell the liver to stop dumping glucose into the blood, because new sugar about to come on board.
Paradoxically, a small meal with some modest carbs (20g does it for me in the morning) will cause an initial bump, but then BG will start to fall and be responsive to insulin.