The prequel series Enterprise eventually explained it as a nasty accident. Some illegally genetically engineered human super soldiers came to their attention when the half dozen of them killed an entire Klingon warship's crew and stole the ship. Impressed, they tried to reverse engineer the super soldier tech from a dead body, but only succeeded in creating and accidentally releasing a virus that made them look more like humans. That affected almost all then-living Klingons before burning out, which is why only original series era Klingons look like yellow peril villains.
Rodenberry apparently said that the Klingons were always meant to look like the modern ones, but it would have bust the budget in the 60s. Nothing to do with rising discomfort with the evil scheming Klingons being basically white guys in yellowface, no sir. Official in-universe justifications post date his death.
Non-canon material published after the first movie (with the first ridged forehead Klingons) and before the sixth (with the first canon explanation of how peace came about) proposed that the old Klingons were Klingon-human hybrids bred to have a better understanding of humans, specifically for service on the frontier. But they were second class citizens with a militaristic culture distinct from mainstream Klingon society (the original series Klingons are notably differently characterised from later versions), and equipped with a decent navy. I'll let you guess how well that went. After the fighting finished the Empire was no longer capable of facing the Federation so had to go for a peace treaty. The Feds are idealistic enough to think they can teach the Klingons cooperation so they go for it. In many ways that's a better explanation IMO, but by the time the official explanation was written (by Judith and Gar Reeves-Stevens, who were actually old school Star Trek authors retained to try to save Enterprise) there was too much official stuff inconsistent with it, so the virus was born.
