What do we mean when we say it today?
Essentially, this phrase just means that someone will make the most out of life in a situation. How do you open an oyster? By prying it open. You can't just gently touch an oyster with a knife; you have to put some muscle behind it. But we think all that hard work is totally worth it because there's a chance of a huge payoff inside. Not only are they totally delicious—sometimes there's a pearl inside.
Hmm, that sounds familiar. Out in the big world, there are endless possibilities, but you have to work hard to find them. And a pearl isn't just sitting around for the taking, but an oyster is. That means that even if you find an oyster, there's no guarantee that it will pay off. Sometimes oysters have nothing inside (but more oyster).
The lesson? Anyone can pick up an oyster, but only the lucky find a pearl inside. If you're quoting this line, then chances are you're an optimist hoping that the oyster you pick up will be one of the lucky ones that actually contains a beautiful pearl.
The subtext of Pistol threatening Falstaff is gone nowadays. There is no sword or threat in our modern version. Instead, we just like to think that if we're persistent enough, we can find those oysters with pearls anywhere in the world.