Faster particles, more collisions
Temperature is a measure of how fast particles move. So at a higher temperature the reactant particles have more kinetic energy.
Step 1
The temperature rises.
Step 2
The particles gain kinetic energy — they move faster.
Step 3
The number of collisions between them in unit time goes up.
Step 4
So the rate of reaction goes up as well.
A double advantage
Temperature is more powerful than the other factors because it does both things. It raises the number of collisions, and at the same time it raises the energy of each one. So the proportion of collisions carrying enough energy goes up too.
In short
Higher temperature = faster particles = more collisions and harder collisions = a greater rate of reaction.