Hi Alli,
Great question!
If you want to feature specific events or ranges of events on your homepage – perhaps using custom layouts – then you’d really have to do this as a further customization.
You don’t necessarily have to reinvent the wheel completely, though. You could for instance categorize the events you are interested in featuring (though we have a concept of ‘featured events’, at this time our widgets don’t allow you to filter down to just those events, however categories are supported) and use the advanced events list widget or calendar widget to display them.
With the aid of a template override you could change how they are displayed and create something very close to what you see with the site you referenced.
Some further resources that you might find useful:
I hope that helps 🙂