Pearson Product Moment Correlation Coefficient method is used only if variables are linearly correlated.
But if they are linearly correlated, then correlation coefficient $$r=\pm 1$$ only.
Then why we find out r by this method and get something like $r=0.6$?
If two random variables are linearly dependent then you will find $r=\pm 1$. To be linearly dependent means that one is a linear function of the other:
$$Y = a + bX$$
However, you can have a linear correlation without linear dependence, for example
$$Y = a + bX + \epsilon$$
where $\epsilon$ is some other random variable which is independent from $X$. In this case you will find $r^2<1$, and how much less than $1$ depends on the variance of $\epsilon$.