What is the rationale behind Bitcoin mining? I don't understand the logic of "mining", if bitcoins are a digital currency, why do we need a GPU to "make 'em"?
Why do we mine?

We need to all agree on who owns which coins.  If your idea of who owns what is different than mine then how can we trade?

To achieve this, the Bitcoin protocol keeps its ledger of all transactions in a series of blocks, with each block recording a collection of transactions.  Since we don't have a central authority to declare which blocks are valid, we run a 'lottery' instead.

Each miner bundles up all the new transactions he knows about, adds in an extra transaction which creates some coins out of thin air and pays them to the miner, mixes in a random number and hashes the whole thing.  If he gets a hash less than a certain number, he wins and his block is added to the blockchain.  If not, he tries with a different random number, over and over.  Everyone else can verify that his random number with his transactions hashes to a low enough number, so they accept his block.  The network periodically adjusts what this target "low enough number" is, in an attempt to keep the average speed of block generation at about 10 minutes.

Why do we need a GPU, not a CPU?

You generate coins by calculating hashes.  Each hash you calculate has a certain chance of making a block, and earning the associated reward, which is currently 50 coins.  The faster you can calculate hashes, the more chance you have of generating 50 coins.  It just so happens that ATI GPUs have the right kind of hardware in them to calculate sha256 hashes many times faster than an Intel CPU can do it.  That's why people use GPUs instead of CPUs.

The Bitcoin network automatically adjusts the "chance per hash" to try to keep the global rate of generation at one block every ten minutes.

In the beginning, before GPU mining started in earnest and not many people were mining, CPU mining worked just fine.  But as more people joined the network and hash rates became faster, the network difficulty increased to keep the blocks coming along every 10 minutes or so.  As a result, CPU mining became less rewarding per CPU.

Then when it became more common for miners to use GPU hardware to mine because it is so much faster and more energy efficient than CPU mining, the global network hash rate increased still further eventually making it completely useless to mine with a CPU; the expected rewards are less than the power consumed by the CPU.

Maybe one day FPGAs or ASICs will render GPUs useless for mining, like GPUs already did to CPUs.