The average cost curve is u-shaped because costs reduce as you increase the output, up to a certain optimal point. From there, the costs begin rising as you increase the output.

To understand why this happens, you need to know what the average cost is. In economics, there are two types of costs: variable and fixed. Fixed costs are those expenses that remain the same. They include things like rent and bills for utilities. Variable costs keep on changing. Inputs are examples of variable costs. You may need more inputs as the demand increases.

Average cost is defined as the total costs (fixed costs + variable costs) divided by total output. As you increase the output and variable costs, the average cost reduces because the output adds value to the consumer.

Assume that you are in a room full of guests, and you give everyone a bottle of water because they are thirsty. You have paid for the venue, which can host a specific number of people (fixed cost). Your variable cost is the amount that you spend on water bottles. You may be forced to increase the water quantity as the event progresses. Buying more water for the guests helps them quench their thirst. Here, the average cost curve continues to slope downward.

However, there is a limit. After everybody is satisfied, the guests begin to waste the water. At this point, your average costs increase because the additional output doesn’t add any value to the consumer. The average cost curve suddenly starts sloping upward, forming a u-shape.

The average cost is U-shaped because an increase in output increases the returns and reduces the total cost. As the curve continues to slope downwards, it enters a phase of constant returns where the returns and output are at their optimum level. After the constant level, continued increase in output stops yielding any further increments in the returns (diminishing returns) and the costs begin to rise, forcing the curve to start sloping upwards.

It is important to note that the total costs are predominantly affected by variable costs because fixed costs remain unchanged. Fixed average costs slope downwards as the output increases, though. The average variable cost forms a U curve following the principle of variable proportions, which explains the relationship between costs and returns in the short term and long term with changes in output. The average cost curve follows the average variable cost curve.

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.