Looking at the site it seems to serve the purpose of mainly being a visual indicator of where the person is in the page while also trying to serve as a non intrusive page specific sticky menu which quickly helps him navigate to important sections of the page.
However the main problem with sticky menus is that they tend to occupy significant real estate space and obscure content and this design tries to avoid that by keeping the dots as a visual indicator and the on hover showing that they utilize them dots to navigate to specific sections in the page.
To quote this article about sticky menus
DISTRACTING AND INTRUSIVE
If not done carefully, sticky navigation can be distracting. Some
sticky elements get delayed when bouncing back into position as the
user scrolls down the page. Others are so tall or wide that they
dominate the layout and impede access to the content. Navigation
should be easily accessible but should not compete with the content
That said, this might not really be effective in Mobile as the site is not responsive (as Alexey pointed out) but also the dots require the users to hover over them to find what they indicate.