When a monopoly's fixed costs increase, why does the price stay the same and the profit increase?
I believe that you must have made a mistake in typing in this question. When a monopoly’s fixed costs increase, it is true that the price the monopoly charges does not increase. However, it is not correct to say that the firm’s profits increase. Instead, the firm’s profits will actually decrease.
A monopoly determines how much it will produce by finding the quantity where its marginal revenues (how much it gets for selling the last unit produced) are equal to its marginal costs (how much extra it costs to produce the last unit). The monopoly makes that quantity and then charges whatever price the market will pay for that quantity (this is determined by the demand curve). When fixed costs rise, none of these factors change. Fixed costs do not affect marginal cost, so they do not affect the location of the point where marginal revenue equals marginal costs. Fixed costs do not affect demand. Therefore, when fixed costs rise, there is no reason for the monopolist to change their price (or the quantity they are producing).
However, when fixed costs rise, total costs rise. When total costs rise and the price you are selling for stays the same, your profits have to go down. It is not possible to charge the same price, sell the same amount of product, and have your profits increase if it is costing you more to produce your goods.
So, the price will not change because fixed costs do not affect marginal costs or demand. However, profits will drop, not rise, because fixed costs do affect total costs.