You can update the stage of a participant in the Participants module by selecting a participant, clicking its Actions menu (the three dots), and selecting the "Update stage"
A pop-up window will then appear, allowing you to change the stage of a participant.
Make sure to save your changes. You will then see the changes appear in the appropriate column in the participant list view.
The stage of the participant can also be viewed and updated in detailed Participant view, in the Stage tab: