In the beginning of the story (means end of Memento) Leonard wore Jimmy's clothes and there was a paper in Jimmy's clothes.
This was how Leonard met Natalie. Why did Leonard do that?
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We don't know for sure why Leonard takes Jimmy's clothes and his car after killing him.
All definite facts the movie tells us about his motives, is Leonard's line when he does it:
I'd rather be mistaken for a dead guy than a killer.
There are a couple of reasonable theories:
I agree with the sense that pervades all of the Stack Exchange discussions of Memento. There is much here left to inference.
But here is my inference, at least at the moment.
As Lenny was confronting Jimmy, he wondered how he would remember that he had done it. He worried (with excellent accuracy, it seems) that he might mistakenly continue to pursue new suspects after killing the real offender. So Lenny decided to leave himself a reminder that he had killed someone by changing into his victim’s clothes.
But then Jimmy said "Sammy," and Teddy showed up and confirmed that everything was different from what it had seemed. So Lenny decided to commemorate not that he had killed the true offender, but (falsely) that Teddy was the true offender; he recorded Teddy's license plate.
Lenny drove to a tattoo parlor, and began to get a new tattoo. Teddy followed him there, and into the actual workspace, to persuade Lenny to change out of Jimmy's clothes. Teddy might want Lenny to change because dressing as Jimmy might get Lenny killed. But for this purpose, why did Teddy actually enter the room in which Lenny was receiving the tattoo? Why didn't he wait until Lenny emerged? Perhaps Teddy was worried that Lenny would commemorate, with another tattoo, that he had succeeded in killing "John G." To prevent this he would have to interrupt the tattoo already in progress. This is ironic, of course; Lenny is there to commemorate not that he had killed the offender, but that Teddy's license plate was that of the offender. And eventually, Lenny does in fact tattoo himself with the message that he had succeeded in killing "John G." only after killing Teddy.
When Lenny left Natalie's house, he found Teddy waiting in the Jaguar. Teddy asked Lenny how he came to be wearing a designer suit and driving a Jaguar; had he done these things while working for the insurance company? Of course the answer to this question should be "no"; the new clothes and car should suggest to Lenny that there's something he's misunderstanding about his new situation. This conversation thus becomes part of a major theme of the film, that to a very large extent we all fool ourselves all the time, failing to see evidence that leads in a direction that we do not want to be led. (Another example of this obtuseness is the scene in which Natalie appears before Lenny with new bruises on her face, and Lenny flexes his hand, noticing that he seems just to have punched someone; but he fails somehow to link these things.)