Pilate heard Jesus's response and continued the idea of releasing Jesus. But the Judeans were not having it. This intense trial takes place in Preparation Day "about the sixth hour" (19:14). Keep in mind that John was using an approximation (timekeeping at that time was not precise) and that he is using the Roman system of time (based on cultural background), making the trial early/late morning.
Then the Judeans pressured Pilate yet again -- but notice their words. They proclaimed that "whoever makes himself a king speaks against Caesar" (19:12). Not only do they make use of Jesus's kingship, but they are also appealing to Pilate's connection with Caesar, the big king of the Roman world at that time. Then the chief priests proclaimed that they "have no king but Caesar" (19:15). This was the straw that got Pilate to carry on with crucifying Jesus in spite of knowing Jesus's innocence.
Even today, peer pressure where a crowd calls someone to do something wrong (notably, drinking while driving and underage drinking) can be a big problem. The crowd can even create appeals that seem enticing but do not make the wrong thing right. That's why watching out for peer pressure is important. Unfortunately, Pilate (and many others) gave in, but Jesus's death and resurrection that came after the pressure can help counter the guilt from doing wrong after being pressured and empower us to take a stand against bad peer pressure.