How do plants grow, die, and then grow again? For instance, when my plants die during the winter, how do they grow again next year? Does it have something to do with the root system? Or do they even die?
It depends on the type of plant, but basically not all of the plant dies. Plants have evolved a number of strategies for winter* dormancy. These are common ones, but probably not an exhaustive list.
Deciduous trees and bushes simply drop their leaves in the fall, and so may look "dead" to the unskilled eye - though with practice, it's usually easy to distinguish between dead and dormant. Then when the weather warms in the spring, new leaves grow.
Other perennial plants may lose some or all of their top growth, even dying back to ground level, but the roots will be alive, and will start growing when the ground warms.
Still other plants have developed specialized underground structures like bulbs & rhizomes - think daffodils, tulips, irises, and similar. The rest of the plant dies, only to grow again from the bulb when conditions are right.
It's worth noting that most, if not all, of these are used for propagation as well, often naturally, and frequently with a bit of human help. Bulbs and rhizomes multiply: the daffodil bulb you planted a few years ago may now be a dozen bulbs, each of which can be moved to grow new ones. Many perennials can be increased by dividing the root mass into pieces, each of which will become a new plant. And cuttings from many trees & bushes can be induced to form new root systems, and become new plants.
- Or summer, dry season, &c. For simplicity, I'll just say "winter".