In the first part of the quote, Einstein says that the "most beautiful experience we can have is the mysterious." By this, Einstein means that we should embrace mysteries and be excited by them. Once we can make sense of something which was previously mysterious, we are able to understand a little more about the world around us. The mysterious is thus "beautiful" in that it is exciting and intellectually stimulating.
In the next part of the quote, Einstein says that the mysterious is "the fundamental emotion which stands at the cradle of true art and science." In other words, according to Einstein, the impulse or motivation for great art and science is always the unknown. There are many great works of art, for example, which address the mysteries of free will, namely to what extent our actions and our behaviors can be attributed, if at all, to our own free will. Slaughterhouse-Five by Kurt Vonnegut, William Shakespeare's Macbeth, and Paradise Lost by John Milton are all such examples. In a similar way, science begins with questions. Beginning with questions like why some pea plants are wrinkled while others are smooth, Mendel developed theories about genes and inheritance. By asking questions about the constancy of the speed of light, Einstein arrived at his theory of relativity.
In the final part of the quote, Einstein says that anybody who "can no longer wonder, no longer marvel, is as good as dead, and his eyes are dimmed." The implication here is that the sense of wonder we have as children is an essential part of what it means to be human. Without this inherent sense of wonder, we cannot possibly see or appreciate the world fully, and therefore we cannot claim to be fully cognizant or fully alive. Without this sense of wonder, we become somewhat deadened to the world around us.