After Mary Magdalene spoke with Peter and John, they both head for the tomb (cf. John 20:3,4). Although they both ran, John, who may have been younger, outran Peter and arrived at the tomb first. Mary followed them back to the tomb, though at a slower pace. By the time John arrived at the tomb, the women were gone and so were the angels (at least temporarily).
"And he, stooping down and looking in, saw the linen cloths lying there; yet he did not go in" (John 20:5). John, perhaps out of reverence or out of respect for Peter, did not enter the tomb at first. He did look into it but did not enter inside for a closer examination. "Then Simon Peter came, following him, and went into the tomb; and he saw the linen cloths lying there" (John 20:6). The impulsive Peter was not content to merely look from the outside. He entered the tomb and wondered about the things he saw therein.
"And the handkerchief that had been around His head, not lying with the linen cloths, but folded together in a place by itself" (John 20:7). Peter, in thinking that Jesus' body had been moved, would have found these details unusual. After Peter entered the tomb, John also entered, and "he saw and believed" (John 20:8). The wording implies that while Peter tried to comprehend what had happened, it immediately became clear to John. There was (and is) no other explanation than the simple truth that Jesus rose again! The Jews would not have taken the body of Jesus away, thereby giving credence to Jesus' claim that He would rise again the third day. The Roman guards would have done everything within their power to stop any thieves since they would lose their lives if they failed in their duty. Grave robbers would not have taken the time or the care to remove the grave clothes and carefully set them aside.
Although Peter saw more after he entered the tomb than John did in his first glance inside, John saw into the meaning of it all better than Peter. John was the first to believe that Jesus was risen from the dead--without being told and even before he saw Him. The resurrection is the foundation upon which Christianity rests (cf. Rom. 1:4), and "the disciple whom Jesus loved" was the first one to deduce that the Lord was again alive!
"For as yet they did not know the Scripture, that He must rise again from the dead" (John 20:9; cf. Psa. 16:10). Though Jesus had foretold His death and resurrection several times, His followers, like the rest of the Jews in that day, struggled to comprehend the idea (though they would soon catch on, cf. Luke 24:45). The Jews believed that the Messiah would not die (and thus, would not be raised from the dead). This fact has great evidential value in that it demonstrates that the disciples were incapable of fabricating the story of the resurrection since they expected it no more than the unbelieving Jews did! Their faith in Christ failed at the cross and out of despair hope was born anew in this unanticipated development.
"Then the disciples went away again to their own homes" (John 20:10). Peter and John now also head for Bethany (which would explain why they don't run into Mary Magdalene as they leave the tomb) to tell the others.