The depth of destructive meshes (DM) defines the amount of layers those DMs have. Typically, each depth has multiple parts per part, and those are all represented by sockets like a skeletal mesh.
For a visual representation, imagine a DM is a bag filled with marbles.
If the DM has a depth of 1, it’s a single bag filled with some marbles. When the bag is broken, it will release its content of marbles.
If the DM has a depth of 2, it’s a bag of marbles that contains bags filled with marbles. When the first bag is broken, it releases the bags of marbles it contains. If any of those bags are broken, they will release their contents (marbles in this case).
This can go up to 5 layers deep with Apex DMs.
If you are using depth support, marble bags will stick to other marble bags they are next to if they are of the same layer or lower. Using this with world support, you can get a crumbling tower of marble bags that can be broken apart rather than just falling over after the first bag is broken.
Why use depth with DMs? Because you want greater detail from your DMs when they are broken. You can see examples of this in Ark, where rocks brake apart in one go, and Conan Exiles, where parts of the rock are broken and disappear before the whole rock is broken; the difference between 1 layer with no support and 3 with support.
Be warned though, DMs have a limit to the number of parts they can have and can get very costly quickly. For example, a DM with a depth of 3 at 9 slices per depth will have 729 parts; all dynamic, drawn, and calculated every frame after the first is broken.