Minecraft – How to get rid of this black hole in the minecraft world

You can burn them, but it's really dangerous. It can get out of control pretty fast.

UPDATE: Shears should now be used to get rid of leaves. They're 6 times faster than using your bare hands (and 4 times faster than using a sword).

Alternatively, you can use a sword if you don't really care about damaging it:

Leaves can be destroyed faster by using a sword, though swords lose double normal durability when used on leaves. As no other tool yields any increase in removal speed, leaves are best removed with bare hands, or a block or object unaffected by durability, so as not to damage your tools unnecessarily.

Finally, as Jonathan mentionned, the leaves will decay after a certain time if it's not attached to solid blocks (but I don't consider this a 'quick' way to get rid of them):

In Beta, leaves decay according to a simple formula: leaf blocks decay unless they are within 4 blocks of a log block, and connected to the log by other leaf blocks. Thus, harvesting the wood from a tree will cause the leaf blocks to vanish randomly one by one.

Hit them with a sword and you'll be rid of them in no time!

It is also possible to use shears to cut through the web.