Simply, transcriptomics is the study of the transcriptome, or the set of RNA transcripts (fragments) within a cell or cell population. RNA is the molecule that makes copies parts of the DNA, a bit like making a photocopy of a page from a book. Although there are many different kinds of RNA in cells, one type of interest is mRNA or messenger RNA, which is involved with converting the genes encoded in parts of our genomes or DNA into proteins which perform specific tasks.
Transcriptomics studies measure the presence of various RNA transcripts in a cell to understand the levels to which different parts of the DNA are being transcribed (like measuring the number of copies of different pages of a book have been copied). In the case of mRNA, since mRNA is used to make proteins, this can gives an idea of the relative levels to which proteins might be present in a cell (although there is not always a direct correlation between RNA levels and protein presence in a cell).
Image from: https://www.lgmd2ifund.org/science-basics/from-gene-to-protein
Different cell types, tissue types, and disease tissues often have very different RNA expression profiles, even in the same individual. This means that even though almost all the cells that come from you and make up your body have the same genome or genetic sequence, they can have vastly different RNA expression profiles. Again, one way to think of it is having the same book (your genome) being present in different libraries (different cell types or conditions) and having different pages being copied at different levels (RNA expression profile.
Transcriptomic studies therefore give us a deeper understanding of the differences between different cell types and between cells in healthy versus disease conditions.
In the past transcriptomic studies largely involved RNA microarrays, but now RNA sequencing has become increasingly prevalent for transcriptomic work.