We can prove that the given language $L$ is Not CFL using the "Pumping lemma for CFLs". That is formal and correct way to prove this. Since this Question is subjective, and that year GATE was a subjective exam, so All other answers which give some vague informal idea (I have written that informal idea at the end of the answer) behind $L$ being Non-CFL would have been awarded zero marks as that doesn't prove anything.
Pumping lemma for CFLs:
Let $L$ be a CFL. Then there exists some integer constant $P \geq 1$ (Called Pumping length or pumping-lemma constant) such that if $w ∈ L$ with $|w| ≥ P,$ then we can write $w = uvxyz,$ subject to the following conditions:
1. $|vxy| ≤ P.$
2. $vy \neq \in .$
3. For all $i ≥ 0,$ we have $uv^ixy^iz ∈ L.$
i.e. Informally, For every sufficiently large string $w$ in $L,$ We must be able to split it such that it is possible to find at most two short, nearby substrings that we can “pump” $i$ times in tandem, for any non-negative integer $i,$ and the resulting string will still be in that language.
- So, In the Whole String $w$ with $|w| \geq P,$ "Anywhere in this string", "Within At Most $P$ symbols", We must find Two sub-strings(Possibly empty, Not Both Though) such that we can "Pump" both of these sub-strings $i \geq 0$ times in tandem i.e. Both repeated $i \geq 0$ times.
Now, Assume that Given $L$ is CFL, Hence, It will satisfy Pumping lemma for CFL.
So, There must be some positive integer constant (pumping length) $\geq1$ existing for this language. Let it be $P.$
So, Now for every string $w \in L$ whose length is greater than or equal to $P,$ we must have some partition $uvxyz$ satisfying all the above conditions.
So, Let me take the string $1^{P} 0^{P}c1^P0^{P} \in L$, Now Try to split it into five parts $uvxyz$ such that All the Three conditions of pumping lemma must satisfy.
Basically in Pumping lemma for CFL, You want to find $"$at most $P"$ consecutive symbols in the String(anywhere in the String) such that you can find two short sub-strings in that part and Pump those sub-strings in tandem.
But for the above String $1^{P} 0^{P}c1^P0^{P}$, we cannot find any such at most $P$ consecutive symbols anywhere in the string which will satisfy the Pumping lemma conditions. (Hint : Check different possibilities i.e. Take Those at most $P$ consecutive symbols completely in $1^P$ part or completely in $0^P$ part or some in $0^P$ part and some in $1^P$ part or some in $1^P$ part and some in $0^P$ part of the string .. etc.. covering all possible partitions.. )
When you take $vxy$ substring completely in $1^P$ part and however you choose $v,y$ in this part, one thing is certain that when you repeat $v,y$ Zero Times then at least one 1 will be removed from the string and hence the resulting string will not belong to the language because the substring to the left and right of $c$ will not remain the same. Similar logic applies for when you take $vxy$ substring completely in $0^P$ part.
When you take $vxy$ substring between $1's $ and $0's$, however you choose $v,y$ in this part, one thing is certain that when you repeat $v,y$ Zero Times then at least one 1 and at least one 0 will be removed from the string and hence the resulting string will not belong to the language because the substring to the left and right of $c$ will not remain the same.
So, Given language doesn't satisfy Pumping lemma and hence the contradiction. So, $L$ is Not CFL.
For more examples to understand Pumping lemma, I have answered many Questions based on Pumping lemma and You can refer to them in my profile under "All Answers" tab.
The Informal idea behind $L$ being Non-CFL is that in PDA, we have a Stack as an auxiliary memory. And Stack works in LIFO manner. Since we have the language as $w_1cw_2$So, if you want to match the substring to the left of $c$ i.e. $w_1$ with that of to the right of $c $ i.e. $w_2,$ You need to PUSH left substring $w_1$ into the stack and skip $c$ and now You need to match the first symbol of $w_2$ with the first symbol of $w_1$ But the first symbol of $w_1$ is at the bottom of the Stack. So, that's the problem with Stack. Hence, we can informally say that Forward matching cannot be done by Stack or for that matter by PDA.
$\color{Red}{\text{Understand Complete Pumping Lemma of Regular Languages, Crystal Clear:}}$
Here is Complete Pumping Lemma Lecture: https://youtu.be/8cdPjuYbIrU
This Pumping Lemma lecture contains EVERYTHING about Pumping Lemma of Regular Languages, i.e. Proof, Examples, Variations, GATE PYQs, Finding minimum pumping length etc. Watch this lecture for Complete, Correct & Clear Understanding.