This basically just means that a force that doesn't move an object doesn't do any work. The implication of the dot product is that if the movement of the object is perpendicular to the force then the force must have not had any effect on that movement and thus the work is 0.

This is because the original formula depends on both the force and the position change.
Since the force varies, we need to break the displacement into small increments, and calculate the work done by the force over each increment. The smaller the increments, the more accurately we can calculate the work done by the force.

We can then sum up the work done over all the increments to find the total work done by the force over the entire displacement. This is effectively what an integral does - it sums up infinitesimally small increments of a function over a given interval.

The easiest way to apply this formula is when the force directly depends on position (for examplle in a spring). When the force depends on some other variable like time we need to try and express it in terms of position instead and use that in the integral.