I have read in a research paper that a typical image of size 256x256 becomes a vector of dimension 65,536. And hence as a point in 65,536 dimensional space.
I cannot understand this. Please explain.
A 256x256 image consists of 65'536 pixels. Assuming that you use grayscale, each pixel will have a value. That gives you 65'536 values/numbers. If you also define an order in which to put these pixels (for example you first take first row from left to right, then the one below and so on; but any order will do), then you can construct a list with 65'536 values in it. For any image you can construct this list of values and for any such list you can reconstruct the image. If two lists are equal then the corresponding images will be identical.
The list you can interpret as a point in 65'536 dimensional space. All you need to consider is that each number in the list is the coordinate along one of the many dimensions (like you have [x, y] in 2-dimensional space).