1. In corn there are two flowering structures:
-"tassel," bearing male florets (incomplete flowers)
-"ear," bearing female florets

2. A female floret will produce a single viable fruit. Botanically, a fruit is any mature ovary. The fruit in corn is each "kernel."

3. A kernel of corn is a type of fruit, called a caryopsis, in which there is only one seed produced per fruit, and this seed is fused to the inner tissues of the fruitcase. For contrast, a related fruit type is an "achene," for which a good example is a sunflower "seed." As is the case with maize, what folks think is the seed is actually the fruit of sunflower. But, in the case of achenes, while there is also a single seed produced per fruit, in their case the seed is not fused, but is loose inside the fruit case. The result is that if you shake the sunflower fruit you can hear the seed moving about inside.

4. The actual seed of corn (as with any cereal grain) is a structure referred to as an embryo or "germ."

Knowing this fact, you can answer the original question posed. You eat the seeds of corn because they are fused to the fruitcase of the corn fruit. If you note, when you eat corn-on-the-cob, or canned sweet-corn, there are often a number of small "saddle-shaped" structures that separate from each kernel. These are the seeds, or germs, proper. Further, you can do this with corn because when humans consume ears directly, they do so at an immature stage of development of the flowering structure (ear) and fruits (kernels), which makes these materials chewable (and in fact delectable, because of their high soluble-sugar content and succulent state). If you waited until kernels were mature to attempt to consume them, you would find both fruit and seed unpalatable and tough. In fact, as you know, the definition of a vegetable is a plant part that is consumed in a tender, or non-mature stage of development.

A pumpkin and a melon are each fruits, not vegetables. These are harvested when fully mature, at which stage the pumpkin and melon fruits have developed a special tissue known as a a mesocarp, and true seeds are embedded within this mesocarp. The mesocarp is fleshy and sweet, and evidently evolved in order to attract animals to eat the fruit, and spread their seeds far and wide in their droppings as a mechanism for plant dispersal. Because of this, the seeds are tought and resistant, and we consider them inedible at that stage.