Breaking up with your partner is not pleasant. They may cry, throw things at you, threaten to hurt themselves etc.
Quitting a job is similar.
I think even outside Japan it can be awkward leaving a job if you feel in some way indebted to your company. Now imagine your boss shouting at you, or maybe worse, appealing to your obligations not to leave the company, convincing you how doing so would make you unworthy of being considered a decent person, or even threatening to give you bad references, etc.
Sometimes it's actually that the other party isn't even malicious and doesn't
have to be as ugly as described above. They may truly believe it's in your best interest not to abandon something.
Many people give in when faced with back pressure in those situations or at least struggle with going through such a stressful situation, i.e. many people know they may not be strong enough to go through with it when facing confrontation, so is it a surprise that a service that acts on your behalf is something people would use?
Now add to that that in Japan people generally try to avoid confrontation (esp. those likely to suffer at the workplace / school) and then imagine maybe being in a company that bullied you for years and you've seen others struggle with mustering up the courage to leave. Also, as far as I know, job hopping is still not seen as good. Many people still hope to stay at the first company they got into for a long time, with parents and partners hoping the same (and often pushing back against job changes).