It is also useful if you are playing a peaceful game.
If you notice an army building on your borders and you do not want to fight them, denouncing them will mostly deter them from attacking...if you have a good reputation.
Remember though, it works both ways. If you are denounced, then attacking them will cause a reputation loss among other leaders, which will cause them to denounce you. You also get a rep gain if denounced, and the denouncement runs out without you going to war.
(Look at the reputation percentage, the higher it is, the less likely other civs are going to turn on you).
The percentage governs how likely neutral parties will declare war on you or friends back stab you (though another mechanism governs that based on many factors - like friends coveting lands, built wonders they wanted, strength of your army, and your reputation level.) Though a civ with a low score in reputation will likely attack, most of the time, they try and remain honorable - or the AI cheats and gives them bonuses.
Denouncing before going to war is a double edged sword though. Having a common enemy and fighting them may give you bonuses with individual civs, but will affect your relations (the reputation percentage) with others who you are not allied with - which affects all sorts of trade agreement prices. But more critically, at some time you will miss step with your allies, which means they will not want to make another declaration of friendship, and then you can lose a lot of goodwill with them, which can cause them to become enemies rather quickly.
Think of it this way. How other civs interact with you is based on your reputation percentage - which is a cumulation of denouncements and war activities (and annoyingly how quickly you sue for peace - which at deity level when everyone demands everything from you at the beginning of the game can cause issues) multiplied by the good and bad things which you have in common with other civs.