How are the scaling parameters included (extracted from) an essential matrix? An essential matrix relates corresponding points between two images assuming that a camera satisfies the pinhole camera model, and can be expressed as:
$$E=K'^TFK$$

where K is the calibration and F is the fundamental matrix.

There are many tools to compute the essential matrix, one of them using OpenCV's function findEssentialMat. You are typically returned a 3x3 matrix which contains the transformation (or the $$E=R[t]_x$$ camera pose version). Typically, the translation is directly extracted from the t part, and the rotation matrix has more than one solution.

However, how is scaling represented in the essential matrix?

In case it is, how can one extract it? In case it isn't, how come it isn't> And how should one include it to properly state the total motion between the images' points?
The Essential matrix is defined only up to scale, so you cannot extract scale from it. In other words, if you multiply $t$ and all the 3D world points in your scene by a constant factor, the essential matrix will be the same.

If you have to get the scale, then you need some additional information. Either you need to have an object of a known size in the scene, or you need some other sensor that tells you what the translation between the cameras is (e.g. GPS, odometer, or IMU). Here is an example where you get scale from a sphere of a known radius that happens to be in the scene.