It seems at this point beyond reasonable doubt that, yes, Jebediah Kerman really did land on the Mun. As the actions of numerous overseers of the Kerbal Space Program have shown, any idiot can land a rocket on the Mun; it's a simple matter of implementation of navball and a basic understanding of orbit intersections (and, perhaps, a slightly-less than suicidal approach to suicide burns).
No, the real conspiracy is in whether Jebediah really made it back to Kerbin in the same rocket. As everyone will know, the official excuse given as to Jebediah's unexpected 200-day extension of his trip to the Mun was that he needed to "relax and soak up the solar radiation" after a stressful flight and landing. Entirely coincidentally, four unkerballed rockets, ostensibly launched to conduct surveys of the Mun and Minmus, were all subsequently lost in the course of their munar expeditions. While we have identified three of these crashsites, it is notable that the fourth vanished without trace - and, a few days after its disappearance, Jebediah decided to expedite his expedition and extirpate any existing worries as to his location, safely splashing down on Kerbin some time later. So in response to your question, I'd contend that Jebediah undoubtedly landed on the Mun- indeed, mirrors he placed there for use in triangulation are still employed today. But the reason for his delayed return? It's quite possibly the worst-disguised incompetence in any extraterrestrial program on Kerbin to date.
This questions is surprisingly complicated. Jebediah Kerman did indeed set foot on the surface of the Mün, but all of the footage you have seen was staged before his mission.
To really understand the chain of events, we have to look at the entire Kerpollo program and its various failures learning experiences, and the political tension between the United States of Kerbia and the UKKR.
After Kerpollo 1 through 10 failed to reach the Mün (due to failure to include engines in the design, triggering the stage separation prior to triggering the engines, deploying parachutes before leaving the atmosphere, pointing the engines the wrong way, launching straight into a solar orbit, mistaking Minmus for the Mün and most tragically, not including enough snacks in the command module) the KSA finally managed to design the Kerpollo 11 mission. Kerpollo 11's launch has been criticized for its rush to launch, because of fears that the UKKR's K1 rocket would put a Kerbal onto the surface of the Mün first. However, the mission made it into orbit safely, and successfully executed the burn for a lunar transfer orbit.
Already the most successful mission yet launched by Kerbalkind, many observers were, perhaps naively, confident that all was going to go well. It's at this point that the "official" story and the real events of Kerpollo 11 diverge. Officially, the ship entered a stable orbit around the Mün, the lander reached the surface and Jebediah Kerman made his historic speech: "óitnim em sedetsu ,oseuq le se on etse!" The videos and pictures you have seen were clipped from the earlier accidental landing on Minmus.
The actual events were far more in keeping with both KSA's trend of rocket quality, and Jebediah's true Kerbal bravery.
When the crew of Kerpollo 11 triggered the separation of the insertion stage, the return stage and lander were also jettisoned. On a safe return trajectory, the crew was set to safely return to Kerbin, but Jebediah was absolutely set on being the first Kerbal to set foot on the moon. Despite warnings from Kouston that while the lander had enough fuel to hit the Mün, it did not have enough fuel to safely land, Jebediah exited the command module, transferred to the Egret, and began his burn to descend onto the münnar surface. Bill and Bob Remained in the command module, and continued their journey home, cheering on Jeb and eating his remaining share of snacks.
Kouston's warnings to Jeb that he did not actually have enough fuel to land safely were correct, and Jeb began planning his contact with the surface. He quickly realized that the ship lacked fuel for a controlled landing, and lacked the structural integrity for ablative lithobraking, and so he decided to make sure that regardless of the ship, he would plant the flag, and set his feet on the surface. Travelling at about 800 meters per second, Jebediah took the flag and exited the Egret, informing Kouston via his last clear verbal radio transmission that he planned to ensure the UKKR would not win the space race. The remaining information of his descent was gleaned from the telemetry relayed to the command module, from his suit.
Jebediah Kerman slammed into the surface of the Mün, traveling just over a kilometer every second, feet first.