Why do protons need to moving at high speed to collide? I want to know Why protons need to moving at high speed to collide?
I know that both protons are positively charged so the electrostatic repulsive force would tend to make them miss each other at lower velocities, but I don't think this really answers my question.
 A: You are right, to get two protons to collide "head on" hard enough so that the quarks inside them can interact requires tremendous force. To see this (here's your homework!) take the equation for the electrostatic force between two (identically charged) objects and use for the distance between them the diameter of a proton, and solve. The answer will give you an estimate of the lower bound for the required force.
A: First of all, it is important to understand that by saying colliding protons, you are not saying that you are colliding them like two balls. Particle collision means that particles scatter from one another. Or in other words, make them interact. Or even in the words of the quantum field theory, make them exchange a particle, which is in the case of two protons typically a photon (a carrier of the electromagnetic interaction). 
Now, you do not need them to have high energies to scatter. The Coulomb force has an infinite range ($F = \frac{e^2}{4\pi\epsilon_0r^2}$). However, really interesting things happen when they get close together. 
For example, one proton could undergo a beta plus decay and transform into a neutron emitting a positron (anti-electron) and an electron neutrino. Further, proton and neutron would bound together and form a Deuterium nucleus ($_1^2D$ Hydrogen atom isotope). To make this happen you need to make them come close enough around the scale of the Deuterium nucles radius $r \backsim 2\cdot 10^{-15}m$. 
