The purpose of single cell transcriptomics is to determine what genes are being expressed in each cell. The transcriptome is often used to quantify the gene expression instead of the proteome because of the difficulty currently associated with amplifying protein levels.
There are three major reasons gene expression has been studied using this technique: to study gene dynamics, RNA splicing, and cell typing. Gene dynamics are usually studied to determine what changes in gene expression affect different cell characteristics. For example, this type of transcriptomic analysis has often been used to study embryonic development. RNA splicing studies are focused on understanding the regulation of different transcript isoforms. Single cell transcriptomics has also been used for cell typing, where the genes expressed in a cell are used to identify types of cells. The main goal in cell typing is to find a way to determine the identity of cells that don't have known genetic markers.
RNA expression can serve as a proxy for protein abundance. However, protein abundance is governed by the complex interplay between RNA expression and post-transcriptional processes. While more challenging technically, translation can be monitored by ribosome profiling in single cells.