Every generator pays to have a guaranteed connection to the grid. If they have sold power within that connection and it cannot be delivered to the customer at the other end of the country because of a constraint half-way along the ESO (Energy System Operator) has to pay them an amount to cover some of their revenue loss (they only get paid for what is delivered) and then find someone else to cover the lost generation. This is basically the same as an airline having to pay to reroute you because LHR is over capacity. What is interesting is that because the GB market is one wholesale region the flow constraints between different parts of GB are solely on the ESO to handle. This is very different from say the constrain that is imposed by the maximum capacity of NSL (North Sea Link - interkonektor między UK i Norwegią). A generator in NO basically has to 'buy' capacity on the link to sell to someone in GB. As long as there is no fault there are no curtailment payments due, instead it is all reflected in the wholesale price.
One thing to note is most CCGT (Combined-Cycle Gas Turbine czyli elektrownia gazowa) plants are located somewhat closer to their customers, so less likely to face curtailment and such costs. However, even there the geographical demand distribution of GB has changed quite a bit in the last 40 years, so there are still some constraints. My understanding is that due to issues with the new pylon design there is a decent chance that Hinkley Point C (najnowsza budowana elektrownia jądrowa w UK) will be curtailed on a regular basis, at least initially.
As for how much you and I are paying, the 2024 curtailment estimate is expected to be ~£1 billion, this amounts to ~0.4p/kWh, so £10-£15 this year for the typical consumers. This is less than the current CfD levy, and similar to things like the capacity market and balancing services. If we don't expand the grid we might be looking at a 3.4p/kWh surcharge in 2030 (in 2030£, so ~3p/kWh today). This is more substantial, but not catastrophic.