Well, that’s a question with many answers, some of them compatible with each other, some not so much. Ultimately, humans are wildly different from each other so they have different motivations overall.
The first one (not necessarily the most common one though) you already pointed out: it’s about envy, pride, the size of one’s Richard. Anyone you see saying anything along the lines of “At your rank, what do you know?”, is inherently driven by this. They think that high rank means more skill and more skill means you are allowed (in their twisted minds), to talk down to others. They think that they’re more knowledgable by the simple fact of them having a higher rank. I like to compare it to old-fashioned sports. I’d rather have an overweight football coach who may not have played a single professional game in his life, but studied a lot and knows what to do than Wayne Rooney or Cristiano Ronaldo, who may both play the game very well (at least at some point in their lives), but show everyone that the might find themselves in deep intellectual troubles if they’re ever in a position where they’re forced to tie their own shoe laces. In other words, if you are heavily reliant on your mechanical skill, you might get relatively far without having a clue about the game. Still, they seek validation by getting better, because getting better means to them that there will be fewer people who are allowed to talk down to them. Live by the sword, die by the sword - because most people won’t be the absolute number one in the world (and those that are usually don’t think that this means they know everything and everyone else knows less - because if they did think that, they wouldn’t have been able to get better and better to reach where they are).
A second option is progress. It’s a bit similar to our first answer, but this one is less toxic. You want to get better, because that’s how we’re supposed to life. You do something, you get better at it. And if you don’t get better, something is wrong. Imagine lifting weights every day and you realize that you can lift less and less each day - that’s problematic, isn’t it? Complacency isn’t healthy, at least not in real life. If you don’t get better at your job, soon you’ll get worse and you might not have it soon thereafter. If you don’t work out issues you have with your partner, because “eh, we’ve been together so long, they know”, it might get bad and the relationship might end. Climbing steadily does not equal “getting better”, but it is an expression of getting better. If you get better (and actually play more than 10 games per season^^), you will climb eventually. Not ever climbing despite trying probably feels bad (to be honest, I never actively tried to climb and failed doing so^^).
The third is the promise of better games. I can still remember low silver. Noone talks, everyone picks DPS, 2 people say they’ll throw if they don’t get Genji, everyone complains about not having healers or tanks but noone swaps, everyone “needs healing”, yet flanks somewhere out there - it can be mighty annoying. After doing the climb, I’m now between mid gold and low plat every season, where almost every time we’d get a healthy 2-2-2, where people are capable of flexing so we can counter stuff and we talk to each other, and almost always in a productive manner. It’s way more fun than silver.
The fourth is the false promise of better games. Climbing doesn’t mean you get fewer throwers, leavers, smurfs, DPS-onetricks, etc. It doesn’t mean that suddendly, everything is fine, but some people still believe. In fact, they believe it so much, it makes them toxic themselves. They’d blame their “silver team mates” instead of being proactive and reducing the chances of a leaver by simply being nice and communicative. They think that everyone else is to blame when things go wrong, because “it’s like that in this rank”. And whenever you blame others instead of yourself, you won’t get better.
The fifth and last I can think of is people who want to play with their friends that are at a higher rank.
Overall, that should cover most bases. Personally, I wanted to climb out of silver and I’m happier for it. Now I’d rather play good games with loads of variety (meaning no comp for the most part), and don’t really feel like climbing much further. As you insinuate: what’s the point, really? I don’t want to stagnate, but I rather get better at my depth than focusing on a specific hero to climb with.