That is basically two steps:
Slideshow Intro
From your INDEX page settings > GALLERY tab, select layout "slideshow", click "use as intro". From there, you will likely want to adjust some settings like "height" and "fit" also. The images that go into your slideshow, are the images uploaded into your index page gallery.
Links to galleries
This is basically the "folders" module, showing links to all galleries inside a folder. To achieve this, you need to decide upon a folder that contains all the galleries you want to display here. Go to INDEX page settings > FOLDERS tab, scroll down and open "Advanced", then select the folder from "Folders Assets":
You can also choose to add an amount to "limit folders", to limit the amount of galleries that display from that folder.