If the selling club decides they don't want to sell, an impasse is reached. If they want the full value of the clause, they can insist that the buying club make an offer of the buyout clause + VAT. Obviously, this whacks somewhere in the region of 20% on the bottom line of the deal for the buyer. Still, they pays their money and they gets their man. However, this still requires the selling club to ACCEPT their offer.
If they really want to play hardball, they can simply refuse to deal with the buying club in any way. In this case, it's the player's prerogative to get out of his contract using the clause. It's his contract, after all. The buying club are merely a 3rd party. How does he get out of it? By buying it out. How does he do that? Well, he deposits the value of the release clause with the league himself. Obviously, if this is tens of millions of euros, he's going to find that difficult. Even the handsomely-remunerated Leo Messi would struggle to meet his buyout clause.
So what happens? Well, in practice, the buying club gives it to him. This was the case in the transfers of Javi Martinez to Bayern and Ander Herrera to Manchester United. Athletic Bilbao are stubborn as Donkeys. They refuse to play non-Basques and, as such, refuse to sell their best players, so hard are they to replace. Bilbao thumbed their nose at the offers from both of the titans attempting to buy their players. They even went so far as to tweet their refusal of United's offer!
Both Herrera and Martinez had to deposit the value of their clause with the LFP, essentially turning themselves into free agents in the process – they had 'bought themselves out' of their contracts. Again, however, there are tax implications. When the money required to buy out the contract hits the player's account (or, more probably, that of the company he uses to manage his assets) it's liable to tax. Income tax is an eye-watering 52% for high earners in Spain. This means A) the Spanish Minister of Economy and Finance's eyes spin Looney Tunes-style and become € symbols and B) the buying club has to deposit more than twice the amount of the clause in the players account to pay the tax as well as the clause fee. Franz Beckenbauer once famously declared that Bayern would not get involved in the 'lunatic arms race' of the transfer market. With the Martinez transfer, it seems that they Bayern hierarchy are now prepared to pay what it takes to remain competitive at the top level.
These clauses have never really been tested under European Law, and are simply a matter of practice between clubs and football authorities. At the time of writing, this is an adequate, if simplified, explanation of how the British fan's second favourite bit of contract law operates.