Hi - I would stick my neck out and say it is just because of convenience. So that we can apply them directly without worrying too much about which row to multiply with what and add etc to get an interchange of rows. It could be that these three were most common of the operations required to bring matrices on reduced form.

Essentially, row operations on given a set of vectors (represented by a matrix), is that you do some linear combination so that the linear span of the resultant set of vectors is same as the starting set of vectors.

I do not know if there is any geometric meaning to row-operations.

Will appreciate if anyone can offer more insight here.

Thanks