This discussion revolves around the old observation that still water stagnates. Without intellectual stimulation, a person grows dull. Learning and traveling are what make life pleasurable and exciting. As Robert Browning wrote, "A man's reach should exceed his grasp, else what's a heaven for?" Indeed, it is part of human nature to be inquisitive. After all, only the most primitive seek nothing new.
To be human is to constantly strive for greater knowledge and understanding of our place in the cosmos, in every sense of this phrase. Therefore humanity's natural curiosity does not allow us to be happy with the current boundaries of understanding and knowledge. There is something in us that constantly seeks to discover more knowledge and understand more about the world in which we find ourselves. Leaving the known for the unknown is therefore fulfilling our promise as humans.
One reason that it seems essential to leave the known for the unknown is that it is only bypursuing the unknown that we achieve a greater sense of how much we both do and do not know. Consider, for example, the development of ever-more-sensitive scientific instruments, such as microscopes. We can only know as much about tiny things as our microscopes allow us to see; the more we can enhance our ability to see, the more we are likely to know about things and processes that are often extremely important, even though they are literally invisible to the naked eye. The same is true, of course, of telescopes as well; the more we refine our abilities to see into the universe, the more we are likely to know about such matters as the potential existence of life on other planets, potential dangers from approaching asteroids, etc. etc.
I agree that if we only stick to the known, we can never grow. For example, to this point, we know a certain amount about cancer. To this point, scientists are not able to cure it. The cure to cancer is an unknown. If scientists were to stick to the known, a cure to cancer would never be found. It is only through exploring the unknown that one will (eventually) find a cure to cancer.
This statement seems to be advising us that what we know is not the most important thing, but what is important is what we don't know. It sounds as if the speaker is giving a solid argument for never being satisfied with what we have, but to forever be searching for more information. We can never know everything, and we should be careful not to grow complacent with the knowledge that we have—we must forever be looking forward to discover what lies ahead. We must be searching for the next unknown...and when that becomes known, we continue on to the next yet again...and on.
The best example of this is Outer Space. We have lived our entire existence on the surface of a single world in an infinite universe; it is only within the last two hundred years that we have realistically considered traveling beyond our own atmosphere. With the world's population growing larger every decade, we need to find new resources and new ways to live without draining our own beyond the point of no return. Colonies on the Moon would be the first step in spreading throughout our own Solar System; we leave the Known Earth for the Unknown Universe and so survive as a species.
Often times, the unknown ends up changing what we thought we knew before. Pursuing the unknown has led to every scientific, educational and human advance (although arguably, not every advance has been positive), so this idea is central to all learning, which is a remarkable thought. Pursuit of the unknown is also simply a major, natural human tendency, and it motivates much of what we do in life.
Because if you don't, you never get anywhere. If you cling to what is known, you can never progress. If you, for example, hold on to Mom and don't want to go to Kindergarten, you never start growing up. There are all sorts of examples like that in life. You have to give up security in order to grow as a person.