Why is MC=MR at the profit maximizing level of output?

MC = marginal (extra) cost incurred by a firm when its production raises by one unit.

MR = marginal (extra) revenue a firm receives from producing one extra unit of output.

As a firm is trying to maximise its profits, it needs to consider what happens when it changes its production by one unit.

The firm will of course incur an extra cost from producing an extra unit, but will also receive revenue from that unit.

If the marginal cost is bigger than the marginal revenue obtained, then the firm should realise that producing an extra unit of output was not profitable. The firm should thus cut down some of its production.

If the marginal cost is smaller than the marginal revenue, then it is profitable for the firm to produce an extra unit of output. The firm should continue to raise the production of extra units of output, as long as the marginal revenue it receives from that unit exceeds the marginal cost.

The firm should continue doing this until MC=MR, a point at which they should keep production constant. This is because producing an extra unit beyond this point will create a higher marginal cost for the firm than the marginal revenue it creates (the cost will be greater than the profit).