I know this is the DIY sub, but I would get a few estimates to have the floors redone wholesale. Trying to piecemeal it will almost certainly look worse than leaving it alone. You can rent a floor sander and DIY, but they’re tricky to run with no experience so you’re likely to leave the floor wavy. It’s also fairly expensive. Pros will be in and out in three days. If you do decide to DIY, the finish is applied with a lambswool applicator. blog.cityfloorsupply.com/how-to-use-a-lambswool-a…
If the stains are more than superficial, the options are going to be staining the floors darker yet or removal and replacement of the affected boards. Both of those will require refinishing and the latter is for sure a job for someone with experience. I agree with vodulas, that looks like water or pet damage that has already had an attempt at fixing it. Good luck, those are really pretty floors otherwise.