A monopoly firm is a price-maker simply because the absence of competition from other firms frees the monopoly firm from having to adjust the prices it charges downward in response to the competition. The whole point of a competitive marketplace is that consumers can choose among multiple companies for the same or very similar goods or services. The freedom to choose among competing firms means there is pressure on each firm to offer the best product or service at the best price. Absent that competitive atmosphere, a sole provider can set the price he or she wants.
If demand for the product or service exists, and there is no pressure from competing businesses, the monopoly firm can charge whatever price it desires. This does not, however, mean that consumers will pay any price for the product in question. At some point, a monopoly firm may set prices that consumers calculate exceed the value of the product. In the case of what may be considered "essential" services, the monopoly firm need not worry about exceeding a certain price. This was the case for many decades with the telephone industry. Bell Telephone/AT&T enjoyed a monopoly in the telecommunications industry for much of the twentieth century. There was simply no viable way of incorporating competition into that industry until the advent of personal computers and, later, cellular phones. Once the possibility of competition emerged, the Department of Justice was able to force AT&T to divest itself of much of its empire and allow competition for telephone services.
A monopoly firm can set prices for its goods and/or services in the absence of competition. That freedom to set prices, however, diminishes when the prices in question exceed demand for those goods or services. Unless the product is essential, consumers will simply forgo purchase of the product.