As you enter the Timesheet view you are automatically shown your personal time tracking board. By clicking the "Show person selection" in the right top corner, you should see your name in the search field.

Choose a project and the desired task (either general or a specific task) from the drop-down lists. Only the projects you are added as a resource will be visible. Then simply enter the amount of work hours on the chosen day. Saving the hours is done by clicking enter on your keyboard, clicking the 'Save hours' button or by using the auto-save feature implemented (enabling / disabling this is possible, circled in the screenshot above).

Your hours are now entered in the system. The project & task you used are automatically marked as your favorites; so, the next time you enter this view they will be shown by default.

In the "Select tasks to timesheet" view (top-right corner) you get more control over what tasks and projects are shown for you by default. 

When you have one or few favorites, you can directly enter the hours for your  tasks. You can also quick edit more than one value at once. Click one or few tasks on the desired days (the grey area just right of the edit boxes), which will open a dialog on the bottom of the page ('quick edit the selected timesheet records'). You can either choose a default value, or write your own value and click the 'Save' button.