Why is the average cost curve u-shaped?

The average cost curve is u-shaped (high costs when the
number of units produced is low, decreasing costs as the number of units increases, high
costs again as the number of units gets "too" high) because of two things.  The first is
fixed costs and the second is the law of diminishing
returns.

Most sorts of production have some fixed costs. 
You can't have a bakery, for example, without having a building and equipment like
ovens.  When you first start producing baked goods, the average costs are high because
you are making only a few goods and the fixed cost of the
ovens is high. (High costs/few goods = high average
costs).

As you make more baked goods, the average costs
drop because you are making more units of product in the same ovens.  (High costs/many
goods = lower average costs)

Once you pass a certain point,
the law of diminishing marginal returns sets in and your
variable costs rise.  As the link below
says,

There
are bound to be some inputs which cannot be increased indefinitely, at least in the
short run. When output is high, shortages of these restrict the efficiency with which
such inputs as can be varied contribute to more output. Thus at high levels of output
marginal costs tend to be high, leading to increasing average
costs.

For these reasons the
average cost curve is u-shaped.