I would recommend downloading this zip, which contains the key data files for the whole project and a python jupyter notebook to get a jump start on how to read them, or get started online using this binder, which is exactly the same files, but you can use it online (and then download for yourself if you want).
If you are interested in the raw barcode read counts, you can download the RAWEST data in this zip, or the slightly processed (outliers excluded, and barcode counts combined randomly to reach a maximum of 7 barcodes per mutation (to reduce per-barcode noise)) data that I use for the interactive data viewer in this zip.
These files are formatted specifically for the browser visualizations, so I don't think they are as useful (and are redundant with the stuff above), but here is a file with information on each mutation ("Edge") in the dataset, and this zip contains one data file for each mutation (this information is held in TP_data_by_edge.csv in the binder above, this is just rearranged into one file per mutation).
Other data downloads and additional information can be found through the supplement of the paper.