Why should marginal cost be equal to the price an item is sold at? My interpretation of the condition $P=MC$ is that a firm's cost of producing one additional good should be equal to the firm's price. This means that the next item a the firm produces won't yield any profit to the firm.
What's the point of producing this additional unit? Do you have a hands-on explanation?
You should keep in mind that the definition of profit in economics is not the same as in accounting. In particular, economists always deduct opportunity cost from accounting profits, and the price = MC formula has to be interpreted in this way.

For example, let's say you are a self-employed web-developer and you may make a profit in an accounting sense. Economists will subtract from this accounting profit the opportunity cost of not working elsewhere. Looking at it this way, the profit of a self-employed web-developer is just a wage he pays to himself.

The same is true for larger firms. Those firms usually pay out accounting profits to their shareholders. Economists would interpret this as the cost of using the equity capital provided by owners.