There is a lot going on here, and it illustrates some important topics. Rather than repeat what others have said, I'll note a few things. The geometric point of view is pertinent, and groups have a key role in describing the symmetries of things. I will use $V$ for the special subgroup of order $4$.
First I would note that there is a specific phenomenon here unique to $S_4$ - it is the only symmetric group $S_n$ which has a proper normal subgroup other than $A_n$, and $A_4$ is the only alternating group which is not simple. The existence of this special subgroup $V$ is connected with the fact that a quartic equation is "solvable by radicals" - by taking square roots and cube roots and fourth roots. $V$ is the smallest group which is not cyclic. ($S_3$ is the smallest which is not abelian).
Note also that the elements $(1, 2); (3, 4); (1, 2)(3, 4)$ of $S_4$ together with the identity form a subgroup with the same structure as $V$ - the non-identity elements all have order $2$, and it is somewhat easier to see at first glance what is going on. However the elements here have different cycle-types. The significance of $V$ is that all the non-identity elements have the same cycle type.
Now if you have a collection of all the elements of the same cycle-type in a symmetric group, the group acts on those elements by conjugation. The conjugate of any element is an element with the same cycle-type. You may not yet have studied material on group actions, but a group action on any set with $n$ elements involves a homomorphism from the group to some subgroup of $S_n$. Here, seeing precisely three elements of the same cycle-type, we know there is a homomorphism from $S_4$ to a subgroup of $S_3$ induced by this action - the action of conjugation permutes the three non-identity elements of $V$.
It so happens that $S_4$ has order $24$ and the only elements of $S_4$ which fix $V$ pointwise by conjugation are the elements of $V$. $V$ has order $4$ and is the kernel of the homomorphism. The order of the image is $24/4=6$ and must therefore be the whole of $S_3$.
To go back to the question and how permuting the indices permutes the partitions, any subgroup of $S_4$ rearranges the underlying symbols - let's call them $a, b, c, d$ because numbers sometimes get confusing here.
Think about what happens to $(a, b)(c, d)$ when things are rearranged. If, for example $a$ and $b$ go to $b$ and $d$, then $c$ and $d$ must go to $a$ and $c$ in some order, and you get $(b, d)(a, c)$ of the same type.
The geometric point of view makes this obvious in the pairs of opposite sides of a tetrahedron.
How did this first get noticed? - Well probably many different ways by many different people. I mentioned the solution of equations by radicals because that is one of the reasons group theory got off the ground - with people studying the permutations of roots of polynomial equations.
The significance of $V$ is both that it is small and exceptional (and therefore interesting) and also that it is a first illustration of more general ideas (eg group actions, normal subgroups)