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Essentially BDD is a rough measure of the age of Bitcoins in circulation. The more "older" Bitcoins participate in transactions, the higher is BDD. You can imagine that Bitcoins "age" (accumulate days) as they are sitting in an address and not used. Once they re-enter circulation, meaning get sent from one address to another, they become "new" again. Those days that were accumulating while they were sitting doing nothing are "destroyed". Whence the name - Bitcoin Days Destroyed.
For instance, if you sent @DunceBucket_ 100 BTC that you've got from @CoinGuy yesterday, that's 100 BDD (100 BTC x 1 Day). Now, if you sent me 1 BTC that you've got from @DrWe 100 days ago, that's also 100 BDD (1 BTC x 100 Days). So, movement of older Bitcoins contributes higher value to the overall BDD metric than movement of newer Bitcoins.
While the metric is not perfect it helps to understand if people are hoarding old Bitcoins or use them, which can be indirectly used as an indicator of Bitcoin "economic" activity. It also helps to somewhat filter out people spamming network with tons of micro transactions, sending new Bitcoins to themselves.
If this explanation doesn't work, here is a dialog I found on bitcointalk.org that pretty much nails it:
Okay how are days destroyed?
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