Universal Clipboard is one of those features that "just works" as long as it's set up on all your devices correctly. Here are the requirements needed to use this feature:
Sign into each device with an Apple ID:
Each device you want to work with a universal clipboard must be signed into the same iCloud account. Check these on your Mac by going to System Preferences > Apple ID and Settings > "Your Name" on your iPhone.
Bluetooth: Each device must have Bluetooth turned on.
Wi-Fi: Each device must be connected to the same Wi-Fi network.
Handoff: Each device must have Handoff enabled. On a Mac, you will find this under System Preferences > General:
On an iPhone or iPad you will find this in Settings > General > AirPlay & Handoff:
Generally speaking, this feature "just works" assuming your settings are correct. Keep in mind if you copy a large video or image file, it may take a minute or two to fully copy. Also, if you copy one item but don't paste it immediately on a different device, the clipboard on the device you are using may revert back to what was previously on it.