With wide-field phased array feed technology, the Australian Square Kilometre Array Pathfinder (ASKAP) is ideally suited to search for seemingly rare radio transient sources that are difficult to discover previous-generation narrow-field telescopes. The Commensal Real-time ASKAP Fast Transient (CRAFT) Survey Science Project has developed instrumentation to continuously search for fast radio transients (duration $\lesssim$ 1 s) with ASKAP, with a particular focus on finding and localising fast radio bursts (FRBs). Since 2018, the CRAFT survey has been searching for FRBs and other fast transients by incoherently adding the intensities received by individual ASKAP antennas, and then correcting for the impact of frequency dispersion on these short-duration signals in the resultant incoherent sum (ICS) in real time. This low-latency detection enables the triggering of voltage buffers, which facilitates the localisation of the transient source and the study of spectro-polarimetric properties at high time resolution. Here we report the sample of 43 FRBs discovered in this CRAFT/ICS survey to date. This includes 22 FRBs that had not previously been reported: 16 FRBs localised by ASKAP to $\lesssim 1$ arcsec and 6 FRBs localised to $\sim 10$ arcmin. Of the new arcsecond-localised FRBs, we have identified and characterised host galaxies (and measured redshifts) for 11. The median of all 30 measured host redshifts from the survey to date is $z=0.23$. We summarise results from the searches, in particular those contributing to our understanding of the burst progenitors and emission mechanisms, and on the use of bursts as probes of intervening media. We conclude by foreshadowing future FRB surveys with ASKAP using a coherent detection system that is currently being commissioned. This will increase the burst detection rate by a factor of approximately ten and also the distance to which ASKAP can localise FRBs.