The ending of X-Men Origins: Wolverine strips Logan of his memory so that he can arrive in X-Men (2000) with no past. After Stryker shoots him in the head with adamantium bullets, Logan no longer recognizes Kayla's body and leaves the ruined facility alone. The captive mutants he helped free get away safely, and Stryker is turned aside by Kayla's power before she dies.
Why does it matter? Because X-Men (2000) opens its Logan story with a drifter in Canada who knows nothing about himself, and the 2009 film exists to explain that blank. The final shot, with Logan walking away from the facility, is the hand off. In my opinion, the memory wipe is the film's neatest piece of retconning, since it lets Gavin Hood and writers David Benioff and Skip Woods give Logan a century of wars, a half brother and a lost love while the older film stays intact.
There is a continuity cost, and it could be argued that it is a real one. X2 has Logan searching for clues about his own past while Brian Cox plays Stryker, yet here Danny Huston is Stryker and the mid credits scene has him arrested by military police over the killing of General Munson. Scott Summers appears as a teenager played by Tim Pocock, where James Marsden plays him in the first three films. Victor Creed is played by Liev Schreiber, while Tyler Mane plays Sabretooth, the attacker who hunts Logan and Marie in the first film.
The production context explains some of the unevenness. Fox spent $150 million on the film, according to Box Office Mojo. An unfinished workprint also leaked online on March 31, 2009, a month before release, and Fox said the copy had missing effects and temporary sound and music. The film still opened to $85.1 million in the US. Compared with the later entries, I would argue this one is the only Wolverine film built mainly to serve the series around it, which is why it ends on a handover rather than a conclusion.
The ending then hands off cleanly to The Wolverine, which finds Logan alone in the Yukon haunted by visions of Jean Grey. For the larger sequence, our X-Men timeline shows where this film sits, and the Wolverine movies in order page covers the trilogy.