Through thousands and thousands of hours of lab work by thousands and thousands of different labs. The key point to remember is that each lab was looking only at a particular step of the process, and as the process became better and better understood everyone starting analyzing less things in more detail. From all that work comes our current understanding of this masterpiece of nature which is the biochemical process of photosynthesis.

Scientific research, particularly in the life sciences often consists of taking a known concept or idea, and then dividing it into smaller components and understanding those. Then, once those are well understood, divide those and understand the components that make up those. And so on..

well, what do you mean by "cell activity"? There are specific procedures for looking at specific molecules, organelles or pathways.

well you can monitor concentrations of different chemicals by spectrophotometric assays. you can monitor organelles by fluorescence microscopy etc.
What exactly are you talking about? Your questions are so general that it is impossible to answer them.

If you are talking about certain cellular processes or pathways, one common mechanism is to inhibit or knock out certain stage of the pathway and see what happens. Eliminate some receptor with an antibody or make a knock-out organism that lacks a gene for, say, some enzyme, and see what process gets modified/prevented, and then study the mechanism and function further.

Also, cytoplasmic or secreted compounds can be analysed, their structure, concentration, time point of expression and so forth.

Furthermore, nowadays genetic or genomic expression profiles are becoming very popular (i.e. gene chip analyses), where the expression of many linked genes can be detected and analysed.

So, tools that are commonly used, in addition to what MrMistery said, include many of the standard methods of cellular and molecular biology, including PCR, different blottings, monocolonal antibody staining/blocking, knock-out and/or transgenic organisms (cells, bacteria, plants, animals, fungi), gene chips, microscopy (electron, confocal, standard bright field) et cetera.