The NHS currently pays well below the full market price for many of the drugs it purchases. (That's because the NHS uses it's bulk-buying power to negotiate hefty discounts with drug manufacturers). Further, some other countries use the prices that the NHS pays as a reference point and refuse to pay more for their own purchases. (So the UK's system is effectively reducing the money that drug manufacturers get from selling to those other countries as well).
The USA regards the system as unfair to US drug manufacturers and is likely to make it a condition of any free trade agreement between the UK and the USA that the NHS will in future pay the full market price for any drugs purchased from the USA. In some cases that could mean that the NHS will have to pay several times what they're currently paying. (The idea that the USA will make such a demand isn't something simply pulled out of thin air by those opposed to Brexit. The USA has already included such a condition within its free trade agreement with South Korea).
So, despite the references to 'selling the NHS to the USA' that some politicians (and some elements of the media) like to use, the actual concern is that the NHS will have to hand over far more money to US drug companies. The Labour party has put the figure as an additional burden of half a billion pounds per year upon the NHS but others challenge their figures.
https://www.bbc.co.uk/news/health-50295231