For most of their history, Klingons have had ridged foreheads. (This is confirmed in the episode Star Trek: The Next Generation: Rightful Heir (#6.23)) According to the episodes Enterprise: Affliction (#4.15) and Enterprise: Divergence (#4.16) for roughly 200 years some Klingons and their descendents were infected with a virus which removed the cranial ridges, giving the Klingons a human-like appearance. This was offered as a possible "in-universe" explanation for why, in Trek's early years, the Klingons look like humans, but as makeup became more advanced in recent years, they took on their more familiar look. There are a few different ways to look at this:
1) It could simply be that the Klingons we see in "Star Trek Into Darkness" weren't descended from infected Klingons, and thus never lost their ridges. This is very well possible, since these Klingons seem to live in a very secluded area on Qo'noS (Kronos), and may have never gotten into direct contact with their infected brethren. While ridge-less Klingons ascended to many high-profile positions--none of the Klingons we see on the original series have the ridges--many of Klingons with the traditional look may have remained. In one of the movie's behind-the-scenes documentaries, the design team comments that the uninhabited area of Qo'nos depicted in the movie was intended to be devastated by years of nuclear wars, implying that the Klingons are at constant war with each other. This makes it plausible that this group of Klingons never contracted the disease from their afflicted rivals. It may also mean that Khan's and Starfleet's altercation was regarded as a local incident, explaining why it did not lead to the full-scale war with all Klingons that Marcus had predicted.
2) It's possible that, due to the events of Star Trek which created an alternate timeline, the Klingon Empire developed a cure for the augment virus earlier than they otherwise would have. The divergence has already had big consequences. For example, both the Klingons and Starfleet lost a substantial number of ships due to Nero's attacks, causing both parties to more aggressively look for new ways to regain their strength. Also, the Klingons had captured Nero, and had several decades to study his ship from the future, the Narada, which may have given them several technical advantages. It can be seen in the movie that a moon (possibly Praxis) of the Klingon homeworld has already been destroyed in 2259, something that didn't happen in the prime universe until the year 2293. Admiral Marcus also mentions how the Klingons have aggressively expanded their territory in response to the temporal change. Perhaps this caused them to already find the cure for their ridged foreheads, years before the same event happened in the prime universe.
3) It's possible that these Klingons surgically added facial ridges as a way to preserve the look of the past. This was suggested by the Klingon Antaak in "Divergence."
4) One explanation that is not in-universe is that, as special effects technology and makeup evolved over the years, the creators have the right to update the looks of ships and aliens without any "in-universe" explanation. Many fans disregard the augment virus as an explanation for the human-like Klingon appearance in Star Trek, stating that adding ridges was simply a continuity error derived from an artistic choice by Gene Roddenberry, and not a narrative choice. However, for better or worse, the discrepancy had already been addressed in-universe, in the 'Deep Space Nine' episode Star Trek: Deep Space Nine: Trials and Tribble-ations (#5.6): during a time travel, the crew notices some smooth-headed Klingons and ask Worf about the reason; Worf acknowledges the difference, but implies it is a Klingon matter not discussed with outsiders. The timing and tenor of this line is interpreted by some to be a joke, but it's still canon.
Gene Roddenberry didn't include any type of explanation when the updated Klingon look was debuted in Star Trek: The Motion Picture, but he was known to be a revisionist who retrospectively considered some parts of 'The Original Series' and the feature films as not part of the Star Trek canon. This does not account for elements introduced by other writers and producers, which often clashed with previously established canon, or Roddenberry's ideas. Nevertheless, the "Enterprise" writers have indicated that, if one takes the view that Klingons have always had ridges, the episodes "Affliction" and "Divergence" don't necessarily refute it. It's possible that the changes in those episodes were quickly rectified later, leaving no long-lasting physical changes in Klingon society.