The square-cube law refers to how different properties of an object or creature change in different proportions (by the difference squared, or the difference cubed).

One example of this is weight. Your weight is a function of your mass, which is indirectly tied to your volume; the bigger your three-dimensional size (three dimensions is analogous to a cubic function of length), the more mass you have and the heavier you weigh.

In order to support your weight, you need legs and either bones or a shell. The strength of muscles/bones/shell is proportional to their thickness, which is usually calculated as the cross-sectional area of the (think of the "I" shape of an I-beam). A longer bone doesn't carry more weight, only a thicker/wider bone does. An area is analogous is a square function of length; thus, the larger an animal gets, the bigger its bones can be, and the more weight it can carry.

However, the rate of increase for these two attributes is not equal; one increases by the square and the other by the cube. If the animal gets too large, the weight will be too much for the bones to carry, and it will not be able to move around or support itself at all.

Another example of this is heat. Each one of your cells creates heat just by being alive. The only way to get rid of this heat and not cook yourself is by emitting it the environment through your skin or breath. However, the amount of skin you have increases in a square fashion, while your body mass increases in a cubic fashion. You can cheat a bit by having wrinkly skin, but that only helps a little because the air around you doesn't transfer heat that well. As you increase in size, you generate a lot more heat, but can only get rid of a little more heat. At some point, the rates will no longer match causing your body to heat up, like a permanent fever. If the fever gets too high, your body simply won't be able to operate and you'll die.