Formulae were developed by a continuous scientific process over the time. It is not the work of a single person rather a number of scientists have contributed to the concept of formula writing.
A system of chemical notation was "invented" in 1811 by Jons Jakob Berzelius. The system is based on the "law of definite proportions", stating that all samples of a given chemical compound have the same elemental composition. Joseph Louis Proust actually published a paper stating the law of definite proportions in 1794, but it was not widely accepted until Berzelius added his support to the idea.
Chemical formulae are short-hand representations of compounds using chemical symbols and oxidation numbers. It makes the representation of chemical reactions easier which helps in the study of reactions.