In the last episode, when Kaede got her memories back, Sakuta went out screaming and crying, and his wound started bleeding again. What was the reason of that bleeding?
I haven't read the light novel, so I can only offer speculation based on the anime. From what I can see, Sakuta's chest wounds seem to appear whenever he's faced with a really shitty situation that he can do nothing about.
It was mentioned in the anime that his wounds first showed up when Kaede was getting bullied and had bruises suddenly appear on her body. And the instance that you quote is similar, Kaede reverts to her original self, probably permanently, and there is nothing Sakuta can do about it. He thinks it's all his fault and hates himself for letting it happen.
So my best guess is that the wounds are his version of Puberty Syndrome, and are representative of his self-loathing.
While the speculative answer by @najayaz is excellent, the movie Seishun Buta Yarō wa Yumemiru Shōjo no Yume o Minai (Rascal Does Not Dream of a Dreaming Girl) provides a different explanation. Not surprisingly, it involves Sakuta's first crush Shoko Makinohara — specifically, it ties in with why there are apparently two of them, and why there's apparently a connection between Sakuta's wound opening and the appearance of the older version of Shoko.
The younger Shoko Makinohara will receive a heart transplant in the near future, and that heart will actually belong to Sakuta. Due to Shoko's puberty syndrome, an older version of her exists due to some time-travel / relativity shenanigans. Sakuta's chest wound exists — and opens up whenever the older Shoko is nearby — because of the paradox of two of his hearts existing in close proximity.
How this situation is resolved forms the plot of the movie.