The class progression of 5e D&D is often very linear without a lot of choice. Yes you get to choose a subclass, yes you get to choose spells if you're the right type of spellcaster, yes you get to choose feats and for certain subclasses additional features like a Battlemaster's maneuvers, but still its easy to have a lot of level ups where you are making no real choices, especially after the first few levels, which can feel kind of anticlimactic after waiting months to level up. Multiclassing, or at least having a possible mulitclass in mind, means you are always making a major choice on level up (albeit sometimes still an obvious one) which makes the game more satisfying, for me anyway.
But my main reasons for multiclassing are to create characters who can participate meaningfully in all parts of play, to create new and interesting character concepts, or to fill out party deficiencies (the last usually only as a factor if I'm on the fence).
Generally, other than taking a one level dip in another class to round out the character concept early on, you want five levels of the primary class before you do something else, and maybe 6 or 7 depending on the class and subclass. But eventually most classes have levels that start to look bland or lackluster compared to what is on offer as the starting package for some other class. For full casters with no interest in weapons combat the incentive can be pretty strong to stay pure rather than slow down spell progression, but even then you have the Wizard who realizes that just one level of Cleric could have him healing and blessing and wearing heavy armor, and the only real cost is getting new levels of spells a level later.