You see it in processes where something spreads to its vicinity.
This isn't really a natural example, but if you draw a square on a sheet of graphing paper. Next iteration you fill in each adjacent square. Repeat this process until you get tired of it. The radius increases linearly at a constant rate, but the area, the number of squares, as a function of each iteration, is growing quadratically.
Take a circular forest in a place where there are no fires and no logging. Its rate of growth is proportional to its circumference, which is proportional to its radius. Its area as a function of time is a quadratic function.
Suppose the value of a network to an individual user is proportional to the number of users. Then the total value of the network, summed across all its users, is proportional to the square of the number of users.
