I am trying to understand the homogeneous transformation matrix, for which i don't understand what kind of input it requires.
What bothering me is the subscript (new) used at the "Location of old Origin". What origin do they want me to put in, and why. ?
See the following picture:
I ignored the rotation part, assuming you understand that. The "Location of Old Origin" with subscript "new" means the $x,y$ value of the old origin under the new coordinate system. In the picture, we want to find the new coordinates of the point, which is the old coordinates of it plus the old origin's coordinates under new system. For example, let the old coordinates of the point be $(2,1)$, the new coordinates of the old origin be $(-1,3)$, then the new coordinates of the point is $(2+(-1), 1+3)=(1,4)$.
In this example, you should put $$\begin{pmatrix}-1\\3\end{pmatrix}$$ in the "Location of Old Origin" with subscript "new" section.
The whole formula is for you to find the new coordinates of a point $S$ from its old coordinates. In the above example, $$S_{\text{old}}=\begin{pmatrix}2\\1\end{pmatrix}, S_{\text{new}}=\begin{pmatrix}1\\4\end{pmatrix}$$
If there is rotation with angle $\theta$, the matrix looks like $$\begin{pmatrix}x_{new}\\y_{new}\\1\end{pmatrix}=\begin{pmatrix}\cos \theta&-\sin\theta & -1\\\sin\theta&\cos \theta& 3\\ 0&0&1\end{pmatrix} \begin{pmatrix}x_{old}\\y_{old}\\1\end{pmatrix}$$
Hence, $$x_{new}=\cos \theta x_{old} -\sin\theta y_{old}-1\\ y_{new}=\sin\theta x_{old}+\cos\theta y_{old}+3$$
See the $cos, sin$ part takes care of the rotation, the $-1, +3$ takes care of the translation of the coordinate system.