I use a lot of faucets
That's a very bad idea.
and accumulate a lot of dust.
Correct.
This supposedly increase my transaction fees.
Which is why I said it was a very bad idea. Depending on the "faucets" you are using, it's quite likely that it will cost you more than you receive to use any of it. In other words, a complete waste of time.
Can someone explain in detail, how I can get rid of all this, with the least amount of increased transaction fees?
Load the private keys into a wallet that allows you coin control, and create transactions that bind a single large output with a few small outputs. Wait of the transaction to receive enough confirmations. Repeat.
How quickly you can repeat the process will depend on the size of your large output. Assuming that you have a single 1 BTC output, you might be able to repeat the process ever 12 hours or so if you are lucky. Each transaction can combine about 5 outputs together without a fee.
I use blockchain for the faucet stuff.
You'll need to export the wallet from blockchain and import it into something that gives you better coin control.
I don't get it. What is "dust"?
Extremely small value outputs that use up space in your transactions causing your transaction size (in terms of bytes) to increase and therefore your customary transaction fee to increase.
You'll have a sum of a lot of small transactions in your wallet. Tons of dust = nice block of sand
Incorrect. Distinct outputs are not stored as a single block. They are stored (and spent) as individual outputs. It's a bit like if you open up the physical wallet in your back pocket and pour 10,000 pennies into it. Sure, you have $100 in the wallet, but when you go to spend it you still have to count out all those pennies. They don't just magically melt together and transform into a $100 bill.
so why you wanna get rid of it.
Because adding five separate 10 µBTC outputs to a transaction adds 50 µBTC in total value to the transaction, but requires the transaction to spend an extra 100 µBTC in fees. Therefore, your transaction costs you an extra 50 µBTC more than it would have if you didn't have the dust included in the transaction.
Anyway, couldn't you just send it to an address that doesnt exist?
Sure, but it would cost you money to do so.
Lets say that you have 500 "dust" outputs all valued at about 1 µBTC each. If you try to send them all to 1BitcoinEaterAddressDontSendf59kuE you'd have to pay a transaction fee of nearly 10,000 µBTC (0.01 BTC) just to get rid of the 500 µBTC (0.0005 BTC).