Do I need to print and bring my ESTA to the airport?

You aren’t obliged to bring it with you, but it can be a good idea. When your ESTA form is processed and authorised, you can see this immediately by checking your form. The procedure is normally complete within minutes, but may occasionally take longer. Your authorisation is based on your passport number, and will be marked down in the electronic system so it can be immediately recognised as soon as you present your passport on your check in for your trip and at the time of crossing the border. This is an intentional design by the U.S. government to make the system simple, efficient, paperless, and fully online.
No official document will be sent to you. Your application for travel authorisation will simply be updated to state that it is pending, that is has been approved, or that it has been denied.
The U.S. border control will have the relevant information available to them when you land in the United States, but your airline may not necessarily have access to this, and may choose to request a physical copy of your authorisation. Thus, it would be best to check with your chosen airline to see whether or not you need to bring a print out of the ESTA with you for the flight.
Generally speaking, although it is not essential to print the ESTA authorisation to travel to the USA, it is considered good practice to keep a copy of the printed form showing the granted authorisation for your own records, and for the unlikely event of any emergency. In particular, this is because the form contains your reference number. This reference number is very useful because your ESTA authorisation is valid for 2 years and for all trips to the USA you may complete in such period. However, in order to reuse your ESTA, you must ensure all the information within your file remains correct. If your passport, address, or situation changes, then you should modify your file to renew the authorisation. To do so, you need to visit the website where you originally submitted your application and connect to your file using the reference number you were provided. This is the same as when you make a mistake during the completion of your application form. Thus, it’s a sensible idea to keep the form and your reference number somewhere safe after printing the document.