All of these graphs are in the shape of curves because of the law of diminishing marginal returns. If it were not for this law, the marginal cost, average variable cost, and average cost graphs would be straight.
The law of diminishing marginal returns says that, when at least one input is fixed, a business will eventually experience diminishing marginal returns when it adds more of other sorts of inputs. For example, if I have a restaurant with a fixed amount of cooking space, I can only hire a given number of cooks before my marginal returns drop. After I hire a certain number, the cooks will really not have enough for each of them to do and they will not add as much to the amount my restaurant can produce. Because of this, my marginal costs eventually go up as I try to make more and more of a product. This is why the MC graph is curved.
If the MC graph is curved, then the other graphs must curve as well. Since fixed costs are fixed, marginal costs are determined by variable costs. If my marginal costs fall and then rise, that means that my variable costs are rising and falling as well. This means that the AVC graph must fall and then rise as well.
If variable costs fall and then rise, average costs will do the same. This is because variable costs are one part of average cost and the other part of average costs (which is fixed costs) does not change. If variable costs fall, average costs fall. If variable costs rise, average costs must rise as well.
From all of this, we can see that it is the law of diminishing returns that causes all of these graphs to be in the shape of curves.