How to know when a curve has maximum curvature and why?

Imagine you have a circle that tangent the inside of the curve. If the radius of the circle is small, then it fits, but if the radius gets to be too big, it doesn't fit into a tight bend.

The largest circle the fits the curve at any point is called the osculating or "kissing" circle.

The radius of curvature is the radius of the osculating circle.

Curvature is the reciprocal of the radius of curvature.

Once you have a formula that describes curvature, you find the maximum curvature (or minimum radius) the same way you find the extrema of any smooth function.

Curvature at a point is what it sounds like - a measure of how "curvy" a curve is. How sharply it's bending at that point, if you will. There are many (and well varied) notions of curvature, but by the sounds of your question, this is what you are talking about.

In order to understand your second question, you need to be a bit more precise in telling us what definition of curvature you're using, I think.

EDIT:

From various comments and your edit to your post, my only remark about why the curvature is maximum at the point where the derivative = 0 is that if you imagine the curvature being plotted on a graph (a plot of curvature vs time $t$), there probably be a place where the curvature reaches some local maximum. This will be like a peak - it'll look like the top of a hill. For this reason the derivative with respect to time there will be zero. It depends on the nature of your curve as to whether this is a local maximum or a global maximum.

As you walk along a curve you turn. At any given point, the rate at which you turn (compared to how fast you walk) is the same as if you were going along the circumference of a circle of some radius. The reciprocal of that radius is the curvature. So when walking through a point in the curve where the curvature is $1$, it will feel like a circle of radius $1$, while curvature of $2$ corresponds to a circle with radius $0.5$, and so on. (At least, that is one definition of curvature.)

As for when to find the maximum, differentiating and finding series works the same way it does in any other application: at the point where the curvature is as large as it's going to be, it goes from increasing to decreasing, which means that the derivative goes from positive to negative. It must be $0$ in the middle.