Jesus recently did what the disciples thought would be hard to accomplish -- feeding a crowd of 5000. However, once the big task is done, Jesus told the disciples to collect the leftovers. Once the disciples did so, they found so many leftovers to the point that they needed to fill 12 baskets (6:13).
The leftovers signify two important aspects of Jesus's work. First, remember how Jesus looked to God and did something extraordinary to feed the 5000? The leftovers suggest that whenever Jesus works, he exceeds the call by such a large margin. Jesus doesn't just overcome obstacles; he overpowers through them (compared to barely winning in a close match).
Second, the leftovers that Jesus had (as well as the feeding of 5000 itself) demonstrate that Jesus is the Prophet. For the people, the big feat meant that God was at work. Thus, the 5000 men were now seeing Jesus as Prophet (6:14).