Another explanation that is not as technically accurate, but really clicked on me to understand how it's possible to establish private communication in the open, is the following:
You put your secret message in a box, put a lock on it that only you have the key to, and send it to the other party. They, unable to open it, put on a second lock of their own, and send it back. You remove your lock, leaving theirs, and once again send it to the other party. Finally, they remove their lock too and can open the box without anyone else having had that possibility.
What can also be inferred from this, is how DH is vulnerable to a man-in-the-middle attack. Someone involved in the delivery could pretend to you to be the other party and to them to be you.
You put your secret message in a box, put a lock on it that only you have the key to, and send it to the other party. They, unable to open it, put on a second lock of their own, and send it back. You remove your lock, leaving theirs, and once again send it to the other party. Finally, they remove their lock too and can open the box without anyone else having had that possibility.
What can also be inferred from this, is how DH is vulnerable to a man-in-the-middle attack. Someone involved in the delivery could pretend to you to be the other party and to them to be you.