Here's the illustration that I would use1:
The blue spheres represent blobs of material, where a blob could be any number of atoms or molecules. The red lines represent the bonding force between the blobs. Young's modulus $E$ represents the strength of the bonding force, i.e. the strength of exactly one bond. When you stretch the material, all of the bonds feel the same force and stretch by the same amount (more or less, so to speak, in a vastly oversimplified view of things).
So given that we know how much force it takes to stretch one bond by length $\Delta l$ (that's what $E$ tells us), then how much force does it take to stretch the material as a whole?
Well, first of all we need to count the number of bonds on each layer of the material. The number of bonds per layer is proportional to the cross-sectional area of the material. Twice the area, twice the bonds, twice as hard to stretch the material.
Then we need to consider how many layers there are. That's where $L_0$ comes in. As an example let's say that it takes 1 unit of force to stretch one bond by 1 angstrom. If the material has 100 layers, and you want to stretch it by 100 angstroms, you'll need to apply 1 unit of force. Each of the bonds feels the same force and will stretch by 1 angstrom. One hundred layers, one angstrom per layer, 100 angstroms total stretch.
But if you double $L_0$ so that you start with 200 layers, and you still only stretch the material by 100 angstroms, then each bond only needs to stretch by half an angstrom. And that requires half the force. So double the length, and the force necessary to stretch the material by a given amount is cut in half.
1: the spheres came from here