The release clause (RC) fee is the minimum amount of money that a club has to accept for a particular player. So to answer your question, if Player A has a RC fee of €40M, bidding €39,9M could be rejected if the club doesn't want to sell the player, but they cannot reject €40M, and thus have to give permission for the bidding team to offer a contract to the player in question. A recent example of this is transfer of German talent Mario Götze from B. Dortmund to Bayern FC for €37M. The really interesting bit of it all was the timing of announcement of this transfer as well as the clash between these two rival teams in the Champions League final 2013 on which the Götze did not play due to a hamstring injury.
The main motivation behind having release fee for a player is to give the player some flexibility in case the player is interested in joining another club but the his club is not willing to sell. Having a release clause fee avoids some sort of modern "slavery" and gives players a way out of a unfavourable situation.
From another point of view, it's also a way for the clubs to show roughly how much they value a player. So imagine Lionel Messi and Barcelona; the release fee is around £215M. That's of course an absurd amount of money for a single player, but what that essentially means is that Barcelona FC values Lionel Messi A LOT, he's practically invaluable.
Obviously minimum release fee has some benefits for the player, and to an extent to the club to protect their rights with investment on a player. However more often than not, a club would not be willing to have a minimum release clause on a contract, since not every player is as loyal as Messi, i.e. a player might get frustrated if a club had the minimum release fee to an absurd amount. Some leagues, e.g. La Liga, mandate for all player contracts to have a minimum release fee clause. In other leagues it happens to varying degree.