The Euler method does not take account of the curvature of the solution - so it tends to give different results depending on the step size. RK (depending on the order) does take account of curvature - this makes the estimated "next step" far more accurate. Basically - if you are pretending a straight line is a good approximation of a curve (Euler) you will always be overshooting your solution. But if you take account of the curvature (RK) you can follow the curve.
Compare with the famous hockey quote (Gretzky): Euler skates to where the puck is; Runge-Kutta skates to where the puck is going to be.
I recommend you read about these algorithms - even the wiki articles will be helpful at your level.