In microeconomics, why does price equal marginal revenue?

It is not really correct to say that price equals marginal revenue.  This is only true in limited circumstances.  Specifically, price only equals marginal revenue in perfect competition.

Price equals MR in perfect competition because your demand curve is horizontal.  No matter how much you produce, it always sells at the same price.

In other market structures, you can raise or lower prices.  When you do, MR doesn't equal price.  Example:

Let's say you have a monopoly on something.  You charge $150 for it and no one person buys it.  So now you reduce the price to $138 so more people will buy it.  One person buys it now.  Your price is $138 and so is MR.

Now you reduce the price to $125 to sell more and you sell two.  Price is $125, but what's MR?

MR is actually only $112 while price is $125.

This is because you made $250 selling two for $125 each.  But if price had been $138 you would have sold one.  So you take $250-$138 and you get $112.

So, MR = P but only in perfect competition.