I have tried to find some info across the Internet, but it is really hard, so I am asking here.
Please, take a look at the header here: https://www.softvelopers.com/
How can you achieve the same (or at least be close to) animation? Is it possible to do so with the help of a Responsive Site Designer? This site is built on WordPress, I believe.

Yes, you can do this in RSD or SDv3. You need to give the navigation the position 'fixed', and then in hover state you give its background a wee hint of a background colour. Then there is some delay in the transitions, I believe. Try it, and if you get stuck, just holler!