Basics
Copy the same way as always.
Paste the same way as always.
Press the Main Hotkey (Shift-Command-v, by default) to access your clipboard history. Or just use the Flycut menu (that little red locker, like the one you see above.
The Bezel
The Bezel is what you see when you press the Main Hotkey. It's the semi-transparent box that appears in the middle of your screen, showing you your clipboard history. To get started, just use the up and down arrow keys while holding the command key to step through your history.
If "sticky bezel" option is enabled, there is no need to hold the command key. When you find something you want to paste, press Return and it is pasted while the Bezel closes automatically. If you don't want to paste, just press the Esc key.
Bezel Keystrokes
Here's a complete list of keys you can use in the Bezel.
Down Arrow or Right Arrow: Move backward in your clipboard history.
Up Arrow or Left Arrow: Move forward in your clipboard history.
Return: Paste the clipboard history item currently displayed, closing the Bezel.
Esc: Close the bezel without pasting, preserving bezel position in the history.
Enter (Fn-Return): Move the clipboard history item currently displayed to the front of the history, closing the Bezel.
Command-,: Display the Flycut preferences panel.
Home: Move to the most recent item in your clipboard history.
End: Move to the least recent item in your clipboard history.
PgDn: Move backward ten positions in your clipboard history.
PgUp: Move forward ten positions in your clipboard history.
Backspace or Delete: Delete the clipboard history item currently displayed, closing the Bezel.
1 through 9 and 0: Move to the clipboard history position indicated by the number, or position 10 for zero.
s or S: Save the clipboard history item currently displayed to a file, closing the Bezel. Delete the item from the clipboard history if capital S was pressed. Clipping is saved to a file on the Desktop, akin to how Screen Shots are saved by OS X. e.g. "Desktop/Clipping 2015-07-15 at 08.35.10.txt"
f: Switch between the main clipboard history store and the favorites store. The favorites store appears in the Bezel and works in the same way as the main clipboard history store, aside from having a yellow tint to the Bezel.
F: Save the clipboard history item currently displayed to the favorites store, deleting it from the main clipboard history store and closing the Bezel.
Menu Features
The Flycut menu allows you to select from the most recent items in the main clipboard history store, clear all of them, merge them all into one entry, or access the preferences panel. You can also Option-Click the Flycut menu icon to disable or reenable clipboard tracking, in case you are copying sensitive information such as passwords.
Preferences Panel
The preferences panel allows you to customize a large number of settings, several of which should be documented here but will have to wait for another day.