Can someone explain this in layman's terms?
You may want to get out some graph paper for this one, it honestly helps.
A vector is a mathematical construct that has a magnitude and a direction. Vectors are commonly spoken of as if they originate from a local origin (0, 0) and end at a point (X, Y). So a vector (3, 5) can be thought of as a line from (0, 0) to (3, 5). If you want to isolate just the X component or just the Y component, you can draw a triangle around it by noting an additional point at either (X, 0) or (0, Y); in this case, (3, 0) or (0, 5). The X component of our sample vector is (3, 0) and our Y component is (0, 5).
Think of knockback in Smash 4 as a vector. It has a direction and it has a magnitude. Through calculations, the game determines that the vector is (Kx, Ky). If you don't touch the control stick at all, and disregard gravity, your character will go flying in a straight line to point (Kx, Ky) assuming you started at (0, 0).
But you are holding the control stick for whatever reason. The game notes this and creates a second vector, (Vx, Vy). For the purposes of simplifying this example, Vx and Vy can be assumed to be either 1 or 0. Holding the control stick to the left results in (-1, 0). Holding it to the upper right results in (1, 1). Straight down is (0, -1). And so forth.
When calculating the knockback vector, the game adds this additional vector before finally sending your character flying
along the path of the new calculated vector. So the knockback vector ends up being (Kx + Vx, Ky + Vy). If you're following along on some graph paper, it's as simple as drawing the first vector, then drawing the second vector
starting at the end of the first vector. (3, 5) + (-1, 0) = (2, 5).
The consequence of this is that you can
directly reduce your knockback by
holding the control stick in the opposite direction you are sent flying, thus
subtracting from your original knockback. In situations where you can be sent in an ambiguous direction, such as an up throw or a side throw, you can hold the control stick
at an angle to cover
both options at once. You can also hold the control stick
in the same direction as the knockback in order to
fly further, which can help escape combos.
And this has been your impromptu linear algebra lecture.