I presume it's because they're price makers, but this doesn't really answer much. Furethermore, in a monopoly is it Marginal Cost or Long run marginal cost that's horizontal?
While principles level textbooks do usually assume that MC is constant for the monopolist for simplicity, by no means does it have to be constant. It may indeed be upward-sloping. Also, both the long-run and short-run marginal cost curves may be horizontal and/or curved, depending on the technology in use.
An upward-sloping MC curve will affect the distribution of Consumer Surplus, Producer Surplus and Dead-weight Loss.
The monopolist being a price-maker has nothing to do with the production technology (and hence the cost structure) it faces. The price-making comes from the lack of (real or the threat of) competition.