Walkthrough mode allows you to “play” the progression by clicking on accessible (unlocked) moments, step by step, in sequence. At each step where multiple moments are accessible, you make a choice as to which moment to play. Access to moments is constrained by connections and gameplay gates.
This button activates Walkthrough mode:

To quit out of walkthrough mode, click the stop button. To pause (useful during an automated walkthrough - see below), click the pause button. Click play to resume. You can also step back.

Clicking on a moment in the progression plan “plays” the moment. When a moment is played, the progression state is updated using the progression elements contained in the moment (defined via the moment properties tab).
The moment that has just been played has a blue frame:

The player has just played the “level2” moment.
A “playable” moment has a white background. Once played, a moment’s background colour changes to grey:

But if the moment is replayable (i.e. the Is replayable checkbox in its moment properties is checked), the background colour stays white and the label becomes a little faded. This indicates that the moment has been played at least once and it can be played again.

Moments become playable when their gating conditions, cost or linking conditions (see here to read about links) are satisfied.

The player will have to leave New Zealand and go to Australia before they can adopt a koala.