Why non gaussian variables are independent
The model ICA uses says that there exist some unknown, statistically independent sources, $s_i$ that are non-normally distributed (their distributions are something other than Gaussian):

$$
s_i \sim S(\mu_{s_i}, \sigma^2_{s_i})
$$

where $S$ is some (possibly) known but non-Gaussian distribution with mean $\mu_{s_i}$ and variance $\sigma^2_{s_i}$.

Then it is assumed that what you can actually measure is the addition of these:
$$
\mathbf{x} = \mathbf{A}\mathbf{s}
$$
where $\mathbf{s}$ is the vector of the $s_i, i=1,\ldots,N$, $\mathbf{A}$ is an $N\times N$ matrix (usually assumed to be invertible) and $\mathbf{x}$ is the vector of the actual measurements.

Because the $x_i$ will just be weighted sums of the $s_i$:
$$
x_i = a_{i1} s_1 + a_{i2} s_2 + \cdots + a_{iN} s_N
$$
the central limit theorem says that the distribution of $x_i$ will be closer to Gaussian than the distribution of the $s_i$ (provided certain non-restrictive conditions on the true distribution of the $s_i$ are  met).

Then all ICA really tries to do is to find a $\mathbf{W} = \mathbf{A}^{-1}$ the inverse of $\mathbf{A}$.

There are any number of measures for "non-Gaussianity".  One of the simplest (?) is to use kurtosis as the measure of how far the sample of a random variable is from Gaussianity.

So, to attempt to answer:


  So my question is what is non-Gaussianity here and why its necessary to maximize it to extract the original sources.


There are several different ways one could measure non-Gaussianity. For example, kurtosis can be used as it is known that the kurtosis of a Gaussian is 3. Any distribution with a kurtosis different from 3 is therefore "non-Gaussian" to some extent.

The reason we want non-Gaussianity is because we assumed that the original sources are non-Gaussian.