Even the ESTA application website itself says you will never need to show the data to anyone, so don't worry about it. Many people (including myself) like to be covered so printed a copy, but it's not at all necessary.

You don't even need the reference number - at check in they'll just ask if have you applied successfully for ESTA. Depending on the system they use they might get a 'Yes' or 'No' reply from the DHS system when they enter your passport details. But if you've already entered APIS data (which is likely) these checks will almost certainly have already been done. But they have no access to the ESTA database itself, so the reference number is of no use to the check in agent.

At US immigration they get all the data up on their screens when they swipe your passport.

You will need the reference number if you ever want to update the information at a later date. But even then, if you have lost the reference number you can just apply again (and you'll get a new reference number).