Mary Magdalene went to see the tomb of Jesus -- but the tomb was empty. So she told the disciples that the tomb was empty. From the information, Peter and another disciple went out to check.
The other disciple was faster than Peter and arrived at the tomb first. Then Peter arrived. Both of them saw linen cloths as well as a handkerchief neatly folded. The neat folding of the handkerchief implies that the folding is not a rushed job.
Turns out that the disciple did believe, though neither of them knew that Scripture would say that Jesus would rise from the dead. It's likely that the "believe" is referring to the words of Mary Magdalene. Basically, the disciples knew that what Mary said was right. Then the disciples came back home.