What is the significance of the value of the probability density function at a certain point?

This explanation is very good!

For an object, what is its mass at a certain point? Because a point is infinitely small, the quality of the point must be zero. However, this object is composed of countless points. What if we need to demand its mass? So the concept of density is introduced $$\rho =\underset{V\to 0}{\lim }\frac{△m}{△V}$$, And finally integrate the density to get the mass m.

In the same way, if a point is randomly selected on [0,1] and the probability at a certain point is obtained, the length of the point is infinitely small, and this probability must be 0. At this time, the situation is similar to the above, we need to introduce the probability density p, where $$p=\underset{x\to 0}{\lim }\frac{△p}{△x}$$In this way, we can find the probability that the selected point falls on a certain segment (a, b). Probability $$p={\int }_a^{b}p(t)dt$$

Summary:
Why is it called probability density, because it is essentially the same as the definition of density in physics. Therefore, the function value of this point represents the probability density of this point. The greater the probability density, the greater the probability of a given part of the length.
When we do the questions, there are generally two types.

1. Tell you the probability density function, let you find the distribution function, and integrate it.
2. Tell you the distribution function, let you find the probability density function, just find the derivative.

Just like you do density problems in junior high school physics, there are no more than two types:

1. Telling you the density of objects allows you to find mass.
2. Telling you the mass of the object allows you to find the density.

The probability density function is just as good as the density learned in our physics.