To customize the theme with your own fonts - or just replicate the demo typography, please install and active the Easy Google Fonts plugin.
After you have done so, you can start adding font controls from Settings › Google Fonts. The font controls with the associated css classes that we used in the demo are the following:
After adding the font controls with the associated classes, you can choose the font families from Appearance › Customize › Typography › Theme Typography.
The exact properties that we used in the demo, for each font control are the following: