Packing is definately not something you'll need or want to learn very early in your "career". Like stated, the performance benefit is not as dramatic as one might think (but it is there, no question). But there are so many other aspects you can improve before going there. Besides, it makes the whole experience much more uncomfortable - learning to relax while packed full is an art all by itself. Therefore I would not recommend it at all for recreational diving - it literally takes the fun and enjoyment out of it and why do you dive if it is not fun? But, competitive divers will usually learn it at some point, as it can give that extra edge that may be needed when competing about the final meters and seconds.
However, I'm not going to totally bash packing. Like Naiad said, it's sometimes very useful in getting a full inhale in the first place. Like if your chest is submerged or you're wearing a thick wet suit. Getting a full inhale may be just labouring. But inhaling until comfortable and then packing moderately is much more relaxing.
As to how it is done, simply:
-Suck in air to your mouth
-Close mouth
-Push air into lungs
-Repeat
If you squeeze your nose or use a noseclip, there are only two ways the air can go - to your lungs or your stomach. Play around with it and when you can get it to go "somewhere" without opening your mouth and NOT doing a swallowing movement, you're on the right track.
If you're doing it right, you should start to feel a unmistakeable feeling of pressure building on your chest. Remember to go slow, ie. don't pack to the limit of uncomfort, just until you definately feel it and stop. Then increase it gradually over time.
Also remember, if you increase your packing amount to x and then take a long break, you have to start carefully building up to it again. Many divers have injured them selves by starting to pack huge amounts out of habit, while not being properly stretched to do that...
The kind of "extreme packing" that some WR level divers do, I would recommend against totally. The risks are far greater than the benefits IMO...