As a 4th year OD student, I explain it to my patients like this.
Presbyopia happens as we age. When our eyes look at something in the distance they focus on that object. When we look at something up close, it takes a lot more focusing ability for our eyes to make that near object clear because it is closer to us. As we age, the muscle in the eye responsible for focusing loses its ability to work making it harder to see things up close clearly.
Progressive lenses are a form of multifocal lenses. Whereas traditional bi/trifocal lenses have clear demarcated areas of viewing, progressive lenses start at the top for your distance vision and as you move your eye straight down, the power in the lens progressively lets you focus on objects closer to you. Hence the name, progressives. They are also referred to as no-line bifocals, though they aren't the only type of lens in that category.