The characteristic itself - in this case "Proud of its power" - does not directly change its stat, but rather serves as an indicator of what is changing its stats - Individual Values, aka "IVs".
In case you didn't already know, each stat (Attack, Defence, etc.) can have up to 31 IVs each, and each IV counts as 1 stat point, though to balance it out, these points are given to the Pokemon over time as it levels up to prevent it from being overpowered.
This is where that characteristic comes in. The characteristic is essentially a hint at what your Pokemon's highest IVs are in. Each characteristic indicates one stat, and there are several characteristics for each. Let's take the characteristic "Likes to relax" for example. Since that particular characteristic indicates HP, it means that that Pokemon's highest IVs lie in its HP stat.
Your Croagunk's characteristic, "Proud of its power", belongs to Attack, which means that its highest IVs lie in its Attack stat. This clearly explains things; assuming "highest" also means that that Croagunk's Attack is more toward 31 than 0, we can conclude that the reason it's Attack is higher than the other Croagunk's is that its Attack IVs are simply higher than the other's.
tl;dr: it does not change its stat, but helps in telling you what actually does.