In layman's terms? That is, you understand, considerably more complicated than using the correct mathematical terms.
I'll try! The problem with the simple "Euler's method", use the derivative to project a tangent line and follow that to the next point, is that any error in using a tangent line instead of the curve itself is magnified: not only is the next tangent line, in the next step, starting from a slightly wrong point, but we are using the wrong values in calculating the slope there.
Runge-Kutta, in general, is a "predictor-corrector" method. In a fourth-order Runge-Kutta, in particular, we use the slope at the initial point to "predict" the value at half the step we are using. We calculate the slope at that new point, then go back and average the two slopes. We use that to calculate a new point at the half way value and again calculate the slope there. Using those three slope values, calculate a value at the end of the step and find the slope there. Now we have 4 slope values to use: one at the left end of the step, two in the middle, and one at the right end. Average those 4 values to get a "mean" slope to use for the entire step. Moving forward from our initial point using that "mean" slope gives the next point.