An analogy which can be used to understand the advantages of an asymmetric system is to imagine two people, Alice and Bob, sending a secret message through the public mail.
With a symmetric key system, Alice first puts the secret message in a box, and then padlocks the box using a lock to which she has a key. She then sends the box to Bob through regular mail. When Bob receives the box, he uses an identical copy of Alice's key (which he has somehow obtained previously) to open the box, and reads the message.
In an asymmetric key system, instead of opening the box when he receives it, he simply adds his own personal lock to the box, then returns the box to Alice. Alice uses her key to remove her lock, and returns the box to Bob, with Bob's lock still in place. Finally, Bob then uses his key to remove his lock, and reads the message from Alice.
The critical advantage in an asymmetric key system is that Alice never need send a copy of her key to Bob. This substantially reduces the chance that a third party (perhaps, in the example, an corrupted postal worker) will copy a key while is in transit to Bob, allowing said third party to spy on all future messages sent by Alice. In addition, if Bob were to be careless and allow someone else to copy his key, Alice's messages to Bob will be compromised, but Alice's messages to other people would remain secret.