Why does the marginal cost curve cross the average cost curve at its lowest point?

Marginal cost curve shows the cost to produce an extra unit of output at all output levels.
The average cost curve shows the total cost divided by the output quantity at all output levels.
If the marginal cost is lower than the average cost, the cost of adding one more unit of output is lower than the average unit cost. As a result, it drags down the average unit cost.
This will stop happening as soon at the marginal cost is equal to the average unit cost. Beyond this point, marginal cost will be higher than average cost, which means the cost of adding an extra unit is higher than the average unit cost. This as a result will increase the average cost.
The point where the marginal cost curve crosses the average cost curve is where marginal cost = average unit cost. At this point, average cost will increase no matter if you increase or decrease the output quantity. Therefore, this point is where the average cost is the lowest.