How is $0\cdot\infty= -1$? It is known that the product of slopes of two perpendicular lines is equal to $-1$ ($m_1*m_2=-1$ for $m_1$ and $m_2$ being the slopes of the perpendicular lines $l_1$ and $l_2$). The slope of $x$-axis $=0$; the slope of $y$-axis$=$ undefined (or $\infty$); $x$-axis and $y$-axis are perpendicular to each other. So, it must mean that the product of their slopes (i.e. $0$ and $\infty$) must be equal to $-1$. How is $0*\infty=-1$? Is it really?

As noted in other answers the slope of a vertical line is not defined, so simply we can not speak of the product of the two slopes of the $x$ and the $y$ axis.
If you want you can accord the fact that these two lines are orthogonal with the general rule $m m'=-1$ using a limit. 
Take a line $y=mx$ amd a perpendicular line $y=m'x=-\dfrac{1}{m}x$, than the first line become the $x$ axis if $m \rightarrow 0$ and we find:
$$
\lim _{m\rightarrow 0} m m'=\lim _{m\rightarrow 0} m(-\dfrac{1}{m})=-1
$$
