I don't have the firmest grasp of the process, but sod it, here goes. Correct me where I slip up, anyone.
When you've not eaten for a while, your body relies to a large extent on the breakdown of stored fat (triglycerides) into fatty acids, which are released into the bloodstream then taken up by other cells to be used as fuel. Triglycerides are stored in fat cells (adipocytes). They are broken down into fatty acids in a process known as lipolysis. Insulin is a powerful inhibitor of the lipolysis process - it stops triglycerides being broken into fatty acids and they remain stored in the adipocyte. At the same time, it removes glucose from the bloodstream either through uptake to cells to be burned, or converted into fatty acids through a process known as de novo lipogenesis.
When you eat a meal, insulin is stimulated to a greater or lesser extent depending on both the macronutrient intake and overall calories. Lipolysis is inhibited, and de novo lipogenesis from carbohydrate may occur.
Going from here, I would assume that the more carbohydrate is eaten, the greater the rise in insulin levels, the longer it remains raised, and the longer lipolysis is prevented. The overall effect may then be that the more carbohydrate eaten, the longer it takes to return to fat-burning. I think, but am not sure, that fat eaten during a meal is mostly stored anyway, but with low insulin levels would not hang around for long before being burned in the lipolytic reaction. Using an analogy, when you eat lots of fat it's like pushing a load of people into a room where they can freely exit through a door at the other end. When you add a significant amount of carbohydrate the door at the other end is locked, and the amount of carbohydrate determines how long for.
As for grains vs veg - it comes down to carbohydrate density and accessibility. Grains in the form of flour are 80% carbohydrate, vs around 20% in even the more carb-laden veg, which are mostly water. In vegetables they are also stored in intact cells that must be broken down before the glucose can be absorbed, whereas in refined flours the carbohydrate is almost immediately accessible. Overall, you absorb more carbs, quicker, from grains than veg, so have to produce more insulin more quickly to deal with it.
