The mere term "obligation" speaks to an entity outside of ourselves. If we only always did what we wanted, and followed our inclinations, then this would not be called an "obligation". Rather, to be obliged means to follow a rule or principle that we would sometimes not otherwise follow (because otherwise the obligation would not be worth saying).
However there is a problem that human desires are not singular; we can have multiple contradictory desires. For example, laziness/comfort versus discipline/progress — we cannot achieve both. So there are multiple competing desires.
If we take these multiple competing desires as analogous to personalities, then we can say that doing one thing may serve one personality over the other. This is almost the same as serving one person over the other, crossing into the area of person-to-person relationships. So it's almost the same to say "be nice to yourself" as to say "be nice to another person".
If Kant says "know yourself because it is good for yourself" then he is talking about your higher self because he presumes a lower part of yourself that does not care about this. So we benefit our higher self by following this instruction. If we ignore this instruction, we benefit the lower self.
The next question is: who do we choose to benefit? Why is it better to serve our higher self versus our lower self? Why is it good to be kind and open versus self-serving? This cannot be answered without assumptions, so I will simply give my answer: "consider other people to be the same as yourself". This opens up a broad topic of "how?" which I won't delve into here.
A note on the topic of being perceived as domineering because maybe this was the motivation for the post — one scenario may be that you're an assertive person talking to a non-assertive person; you share your opinion and they do not share theirs, so it may be perceived that you were domineering. The solution has to involve the other person sharing their opinion (either by you encouragement or asking, or by them gaining assertiveness). I don't think it's a productive solution for you to not share your opinion, because then there's even less communication. A proper solution is a resolution, not an avoidance. In this scenario, "know thyself" helps you have more effective communication if you know your motives and can effectively share them. And if you are actually domineering, then "know thyself" means you know that your will is very strong and you either can't control it or you genuinely think it is more important than other peoples'. Whether right or wrong, that's something that's crucial for you to be aware of.