The monopoly firm using a single price-per-unit selects the price that maximises profits. This occurs where marginal revenue is equal to marginal cost, i.e. at the quantity QM with price PM. The producer surplus (profit) the firm earns is the rectangular area CBDF.
However, if the firm switches to two-part pricing, then we first recognise that the firm can charge an up-front fee equal to the consumer surplus, and the consumer would still be willing to purchase the same quantity (Q0). So, the up-front fee could be as large as the area ABC, in which case profits would be the combined area ABDF.
The firm can do even better than that. Profitability is all about creating and capturing value. So, if the firm can create more value (by increasing the consumer surplus), they can capture more profit (by increasing the size of the up-front fee). So, by lowering the price to PS, the consumers would be willing to buy the quantity QS, and would receive consumer surplus equal to the area AEF. By setting the up-front fee equal to AEF and the per-unit price at PS, the firm then increases their profit to be all of the area AEF.
We can also show the effect of two-part pricing using the consumer choice model. This is illustrated in the diagram below. The black budget constraint represents the most the consumer can afford to buy with their income, when there is a single price-per-unit for Good X (with 'All Other Goods' [AOG] on the y-axis). The consumer purchases the bundle of goods E, which includes X0 of Good X, and A0 of All Other Goods.
With two-part pricing, the firm charges an up-front fee (so the budget constraint starts at a lower point on the y-axis, since paying the fee is like giving up income for the consumer), and a lower per-unit price. So the budget constraint for two-part pricing (the red budget constraint) is flatter. Let's assume it passes through the point E (so the consumer could still purchase that bundle of goods if they wanted to. There is one other point that we need to recognise - if the consumer buys none of Good X, then they do not need to pay the fee. So Bundle C is also an option for the consumer.
With two-part pricing, this consumer can now reach a higher indifference curve, by buying the bundle of goods D (their new best affordable choice). This bundle includes more of Good X (X1), and less of All Other Goods (A1). Because they are buying less of All Other Goods, they must be spending more on Good X.
Two-part pricing works well when the firm faces homogeneous demand for its product (i.e. when all consumers have similar demand for the product). We can also use the consumer choice model to demonstrate why two-part pricing doesn't work so well when there is heterogeneous demand.
Consider two consumers - one with low demand (shown on the diagram below with the blue indifference curves), and one with high demand (red indifference curves). With a single price-per-unit, the low demand consumer buys Bundle G, which includes X1 of Good X, and A1 of All Other Goods, and the high demand consumer buys Bundle J, which includes X3 of Good X, and A3 of All Other Goods.
When the firm moves to two-part pricing instead, the low demand consumer can no longer afford bundle G (it is outside the new budget constraint). The highest indifference curve they can get to is I0, where they buy Bundle C. This bundle includes none of Good X. These low demand consumers find themselves better off by not buying any of Good X at all, because then they don't have to pay the up-front fee.
The high demand consumer would be better off moving to buying Bundle K, which is on the highest indifference curve they can now reach. Bundle K contains more of Good X (X4), so two-part pricing does induce these consumer to buy more. However, Bundle K includes more of Good A (A4), which means that even though these high demand consumers are buying more of Good X with two-part pricing, they are actually spending less on Good X.
So, two-part pricing doesn't work so well for heterogeneous demand, because the lowest demand consumers will stop buying the good entirely, while the highest demand consumers will buy more of the good, but spend less on it. The combination of these two effects is likely to reduce the firm's profit.
This is why you don't often see two-part pricing alone 'in the wild'. Most often, firms will price discriminate first (often through menu pricing - offering different options to different consumers, knowing that each option will appeal to a different 'type' of consumer), then within each subgroup use two-part pricing.