Quantum mechanics doesn’t directly cure anything. It explains things that are small, and it just so happens that cystic fibrosis is a disease that originates due to something going wrong on a small level — with a tiny protein called CFTR. CFTR is what’s known as an ‘ion channel membrane protein’, as it’s a protein that sits on the cell’s membrane. It then controls how much salt (specifically chloride ions) is let into (and out of) the cell.
People with cystic fibrosis have a mutation in the gene that contains the information of how to build CFTR, with the end result that far too much salt is removed from the cell — if you licked a person with cystic fibrosis, their skin would taste salty.
There are many reasons why CFTR can be mutated, and there are many things about how the protein functions ordinarily that are not perfectly understood. Quantum mechanics is necessary to describe both how bits of the protein function (although as it’s quite big, classical mechanics mostly works), and people studying the protein may use techniques that rely on quantum mechanics to help them. For example, my partner happens to work on a related problem, trying to look at another ion channel by a clever technique that uses the quantum-mechanical nature of light and dyes to measure distances on that tiny, tiny scale.