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 ?

The meaning is that if the probability density function is continuous at $a$, 
then the probability that the random variable $X$ takes on values in a short interval of length $\Delta$ that contains $a$ is approximately $f_X(a)\Delta$. Thus, for example,
$$P\left\{a - \frac{\Delta}{2} < X < a + \frac{\Delta}{2}\right\} \approx f_X(a)\Delta$$
and the approximation improves as $\Delta \to 0$. In other words,
$$\lim_{\Delta \to 0}\frac{P\left\{a - \frac{\Delta}{2} 
< X < a + \frac{\Delta}{2}\right\}}{\Delta} = f_X(a)$$
at all points $a$ where $f_X(\cdot)$ is continuous.

"it is a measure of how likely it is that the random variable will be near $a$."

If by "near $a$" we mean in the interval $\left(a-\frac{\Delta}{2}, a-\frac{\Delta}{2}\right)$ where $\Delta$ is some fixed small positive number,
then, assuming that $f_X(\cdot)$ is continuous at $a$ and $b$, the
probabilities that  $X$ is near $a$ and near $b$ are respectively
proportional to $f_X(a)$ and $f_X(b)$ respectively, and so the values
of $f_X(\cdot)$ at $a$ and $b$ respectively can be used to compare
these probabilities.  $f_X(a)$ and $f_X(b)$ respectively are a measure 
of how likely $X$ is to be near $a$ and $b$.
