(Please be aware I'm talking about 5.0 here. Everything I say here is wrong for 4.0 and 3.5, and I don't know anything about the new edition.)

It's a little complex.

5e wasn't really designed for multiclassing, the original intention was to disallow multiclassing altogether, and the feature was only added last minute after playtesters complained. They didn't really try to balance it other than trying to make it weak enough that most players would be dissuaded from taking multiclass levels.

The main cost of multiclassing is that it delays your class features for a level, for some classes, such as primary casters, this is a big deal - a wizard doesn't want to wait until level 6 for fireball. For other classes such as rogues and monks, they might not really care as much about higher level features. Once a rogue (played by a player who is minmaxing for power) has reliable talent, they often multiclass into something that'll give them new features (Paladin is quite popular as it increases your damage, gives you a better hit die, and adds new spells and abilities.)

One of the biggest reasons to multiclass, or refrain from multiclass is roleplay - as this is entirely based on a character's narrative and the story being told, the consideration of whether or not to multiclass for story reasons is essentially a personal choice.

For players looking to maximize their power, it's important to consider whether the features you'd be taking in your new class are useful in your campaign. I mentioned Mask of Many Faces, but that's a useless ability in a pure dungeon-crawler style "fight a new monster each week" style campaign.

It's also important to think about the scope of your campaign. Many people look at the level 20 capstone abilities, which can be extremely powerful, and will argue that multiclassing is never good because it locks you out of those abilities - but consider how long you're going to be playing your character and at what levels. Most campaigns that even get to level 20 wrap up after a couple of encounters, so is that exceptional power, for two sessions, better than taking a new feature that will last you for tens of sessions?

Most character classes, especially martials are very front-loaded. The first 5-10 levels contain most of the core features of your class, and before you've unlocked those, it's often tough to justify multiclassing - but once a monk gets access to stunning strike, do they care about other Ki powers? If a campaign is planned to end on an even level, does a full caster care about that last level when they're not getting any new spells?

As a very very rough rule of thumb, full-casters often want to avoid multiclassing altogether, or if they do multiclass, they want to take a single splash level of another class once they have the spells they care about. For most of those classes, that's either level 6 or level 14. Martials are often more flexible, as higher level features often amount to more choice in abilities rather than mechanical power. Swapping to a different class offers different choices, and certain builds can by synergistic.

Rangers always want to multiclass, because ranger sucks. /j