To turn the feature on, look for the “more +” button, and the option to enable subtitles should be there. To have it enabled by default on all future calls, that can be changed via Settings > Calling > Call subtitles.
As it is, the feature automatically scrolls through in your call. Skype says that in the future, you’ll be able to scroll through them in their own side window, among other additional options. There’s no mention of when this particular new feature will roll out. In addition to the live captioning and subtitles, Skype is also rolling out translation support for over 20 languages. This will be added in the coming weeks, but again, no specific date has been given. (Source: Skype [1], [2])