why a density is not a probability

Here is the single most common misunderstanding about continuous random variables. People see a density curve, point at its height f(x) at some value, and say that is the probability of x. It is not. A density is a concentration of probability, not an amount of it. The clearest sign: a density value can be bigger than 1, and probabilities can never exceed 1, so the height simply cannot be a probability.

Think of physical density instead. The density of iron is about 7.9 grams per cubic centimetre, but that number is not a mass; to get an actual mass you multiply by a volume. In the same way, f(x) is probability per unit length, and to get an actual probability you multiply by a length: for a tiny interval of width dx around x, the probability is approximately f(x) dx, and for a real interval [a, b] it is the area integral from a to b of f(x) dx. Height times width gives area, and area is the probability.

Pushing this to its logical end gives a startling but correct fact: for a continuous variable, every single exact value has probability zero, P(X = x) = 0, because a single point has zero width and so zero area. Zero probability does not mean impossible here; it means the chance of hitting that one value exactly, out of a continuum, is negligibly thin. This is why for continuous variables it makes no difference whether you include the endpoints: P(a <= X <= b) equals P(a < X < b).