It was very risky to go to the tomb of Christ especially in those three days, when Roman guards protected it and, in general, since He was considered as an enemy of Rome (that was the title of His guilt: "The [self-appointing and illegitimate] King of Jews", then any of His associate could be also caught and prosecuted. For that reason, the disciples hid themselves in a house with doors closed, due to fear of slanderous Jews and Roman authorities. Therefore, John and Peter show extreme courage when they, and only they from the disciples, run towards the tomb to check whether the stories of women were right.
We can be confident that both Peter and John ran at maximum speed they were able to run; and then question arises: why did not John make his pace slower and run in rhythm of Peter? But this would be totally unnatural and fake discretion and politeness! Who on earth would have thought of any etiquette, discretion, or politeness when such an unspeakable miracle, the Lord's resurrection, was intuited and suspected! Thus, John simply speaks about a bare fact that he, perhaps because of younger age, was running much quicker than Peter. John needed mentioning of this fact in order to show what happened next, that he had enough time to stop at the entrance of the tomb and contemplate. This was enough for him and he entered not even when Peter came and entered. By that John humbly shows that his faith was most stable, because it was based on highest faculty of human essence: mind and contemplation, which is mind's activity. He was loftier in his mind than Peter and Thomas who demanded physical attestation and touch of the intellectually graspable reality of the Resurrection.
Logic of this contemplation must have gone like that: 1. If anybody has stolen the body then there are just 3 possibilities: i) disciples; ii) Jews; iii) Romans. i) possibility is excluded, because the most faithful of the disciples were hiding in fear, so how could they find in themselves such an uncommon valor as to steal the Lord's body from a tomb protected by Romans? 0 possibility! ii) Neither Jews could steal the body for that would be radically against their intent to eliminate any notion of Christ's resurrection in three days; iii) Nor Romans could do it, for they were specially appointed and commissioned to protect the tomb, so to fail the mission would imply a punishment for them, to say nothing to intentionally spoil the mission by stealing the body - again 0 possibility obtains. And with those three 0 possibilities, one still could think that some extremely audacious aficionado of Christ (for instance, Thomas, who was not always with the other 12, or some apostle(s) from the 70 rest of the apostles) dared to move the stone, enter the tomb and steal the body, then he would have done this together with all those wrappings and linens, for he would be altogether a stupid thief to do this difficult and time-consuming thing in the tomb and expose thus himself to the imminent and unavoidable danger!
What remains? For John, there remains the guess and faith of the Resurrection of Lord. Thus, he, John, is more blessed than other apostles, for they have seen Him resurrected by physical eyes and believed, whereas John believed even without seeing Him with physical eyes (later he did also that) in His resurrection. His purpose was to convey exactly this when he emphasized his swift running and his not entering the tomb. But as a possessor of virtue of humility, he does this in a very misty, oblique, shrouded way.