The Mishna in Yoma 8 (9) quoted in your link says
עברות שבין אדם למקום, יום הכפורים מכפר........ עברות שבין אדם לחברו ,
אין יום הכפורים מכפר, עד שירצה את חברו
Yom Kippur atones for sins between one and Hashem but does not atone for sins against one's fellow man until one appeases him.
That means that if I did not appease my fellow-man before last Yom Kippur, my sin against him and Hashem remains un-atoned for and I must appease him now.
This was codified as Halacha by the Rambam הלכות תשובה, פרק ב, הלכה ט
Teshuvah and Yom Kippur only atone for sins between man and God; for
example, a person who ate a forbidden food or engaged in forbidden
sexual relations, and the like. However, sins between man and man; for
example, someone who injures a colleague, curses a colleague, steals
from him, or the like will never be forgiven until he gives his
colleague what he owes him and appeases him.
[It must be emphasized that] even if a person restores the money that
he owes [the person he wronged], he must appease him and ask him
to forgive him. Even if a person only upset a colleague by saying
[certain] things, he must appease him and approach him
[repeatedly] until he forgives him.
And by the Shulchan Oruch Orach Chaim 606
- Sins that are between one and his fellow, Yom Kippur does not atone for unless he appeases him (receives forgiveness). Even if he
only injured him with words, he must attain forgiveness. If he does
not forgive him at first, he must try again and go to him a second and
a third time. On each attempt he should bring with him three men.
And if he does not forgive him after three attempts, he is unable to
force him to do so. [However, he should say afterwards before ten men
that he wants that person to forgive him.] And if that man is his
Rabbi, he must go to him many times until he forgives him. [The
forgiver should not be harsh in forgiving if he does not want to give
in to the requester’s plea. If the requester gave him a bad name
(הוציא עליו שׁם רע), one is not obligated to forgive him.]