The Goblet unexpectedly selects Harry as a fourth champion. Dumbledore tells McGonagall and Snape about Ministry Official Barty's insistence that the champions are bound by a magical contract after being selected. Then, he took something out of his head with the help of his wand and dipped it into the Pensieve. And the name "Harry Potter" showed up:
Why did Dumbledore do this?
Dumbledore is intending to interrogate the memory (of Harry's name emerging from the cup) more closely, to determine how it could have happened that Harry's name came to be in there.
DUMBLEDORE: Curiosity's not a sin Harry, but you should exercise caution. It's a pensieve, useful if like me you find your mind a wee bit stretched. It allows me to see what small things I've already seen. You see Harry I've searched and searched for something, some small detail. Something I might have overlooked, something that would explain why these terrible things have happened. Every time I get close to an answer it slips away, it's maddedning.
In addition to the answer above, this passage from the book is helpful to understand how the Pensieve works.
"I use the Pensieve. One simply siphons the excess thoughts from one's mind, pours them into the basin, and examines them at one's leisure. It becomes easier to spot patterns and links, you understand, when they are in this form." [Dumbledore said]
"You mean. . . that stuff's your thoughts?" Harry said, staring at the swirling white substance in the basin.
"Certainly," said Dumbledore. "Let me show you."
Dumbledore drew his wand out of the inside of his robes and placed the tip into his own silvery hair, near his temple. When he took the wand away, hair seemed to be clinging to it - but then Harry saw that it was in fact a glistening strand of the same strange silvery-white substance that filled the Pensieve. Dumbledore added this fresh thought to the basin, and Harry, astonished, saw his own face swimming around the surface of the bowl.
Harry Potter and the Goblet of Fire - Chapter 30: "The Pensieve"
So although the film is slightly different (the Pensieve is introduced in a different place in the story), Dumbledore is removing the thought from his head using his wand, and adding it to the Pensieve so he can look at it again more closely and try to find links with his other thoughts.
You see the image of Harry's name that came out of the Goblet of Fire, so that's the memory that Dumbledore has just added to the Pensieve to re-examine.
