On the ‘Edit staff’ page you can use the ‘Assigned projects’ tab to assign projects (or un-assign) to this specific team member.
Just choose one and click on the ‘Move’ button:
Or even add all:
If the staff member has already logged hours on some project, you will not be able to move that project to the ‘Unassigned’ column (such projects are gray on the list), as you can see in the screenshot below:
When you save the changes you made, this staff member will appear on the Project page as a project member and he will be able to log hours onto this project.
Assign member on a Project page
You can also assign/un-assign members on the ‘Edit Project’ page (Time&Projects -> Projects -> choose a project and click ‘Edit project’):
In the ‘Staff’ block, pick members from the ‘Select a team member’ dropdown.
Note: you can un-assign someone by clicking the cross icon at the end of the line, but if the member has already logged hours to this project – you will not be able to un-assign him/her.