Fruits and vegetables contain soluble and insoluble fiber in varying proportions. When you eat soluble fiber, it absorbs water to form a gel-like mass, which helps you feel full and provides other health benefits. It prevents a surge in blood sugar after you eat by slowing down the absorption of carbohydrates.
If the blood glucose level falls to dangerous levels (as in very heavy exercise or lack of food for extended periods), the Alpha cells of the pancreas release glucagon, a hormone whose effects on liver cells act to increase blood glucose levels.
The answer is insulin. Our bodies release insulin right before and during eating, and that insulin tells our bodies to start taking glucose out of the blood, thus lowering our blood glucose levels. It does this by both promoting uptake of glucose by cells and the storage of glucose within our cells.