probability density function In a book the following sentence is told about probability density function at point $a$: "it is a measure of how likely it is that the random variable will be near $a$." What is the meaning of this ?

It is not a meaningful sentence.
For a continuous random variable, the value of the probability density function (pdf) at a point tells us nothing about the probability that the random variable lies near that point. This is because the random variable is continuous, so the probability that it evaluates to any given point is of measure zero.
What is meaningful is to talk about the integral of the pdf near the point $a$, as this yields a probability. So a much better sentence would be:
"The definite integral of the probability density function over a small neighbourhood of the point $a$ is a measure of how likely it is that the random variable will be near $a$"
