Why is the marginal cost curve in the short-run u-shaped?

The basic reason for this is the fact that, after a certain
point, returns tend to diminish as output goes up. This is called the law of diminishing
returns.

At first, marginal costs tend to go down as production
increases. This is typically because of specialization in the labor force. For example, if you
have 1 person making shirts, they have to do the whole job. But then if you add another person,
one can cut and the other can sew. If you add another, one can cut sleeves, another cuts the
body, and the third sews. In this way, you actually reduce your marginal costs as you add
output.

At some point, however, the law of diminishing returns sets
in. Once you have the right number of workers, making more shirts ends up costing you more
because you have to pay overtime, for example. This could also happen because you have to hire
another person to cut sleeves but that doesn't really help much because it's the short run and
the number of sewing machines is fixed and the person sewing can't really keep up with all the
cut pieces coming at them.

So, marginal costs will tend to go down
at first as specialization helps you make more, but they will go back up as the law of
diminishing returns kicks in.