I just encountered this issue myself. When I looked at the BigSur installer log it appeared it may not have been able to establish a network connection for some reason. I had both Ethernet & WiFi connected at the same time so I unplugged the Ethernet cable and Option Booted to the "Installer" that is located on "Update" volume of the main boot drive and the install process was able to finish. I looked at the install log after BigSur finished successfully installing and I still saw some network issues listed in the log so I don't really know if the installer couldn't figure out which network to use or if I just got lucky. This was after two days of attempting to install Big Sur.
Boot your Mac normally. If you cannot boot normally, then Option Boot the Mac and select the main boot volume (default name is "Macintosh HD"). Once booted into macOS you can then access the "Update" volume which can be found under "Locations" on the left pane of the Finder window. There should be a file(s) named something like "install log" (look at the most recently dated one if there are more than one log). Scroll to the bottom of the file to see the last page of log entries to see if anything interesting stands out. Maybe post the last page of the log here.
To "resume" the aborted install process you can try Option Booting and selecting the icon called "Installer".
If none of this helps, try a PRAM Reset (hold the PRAM reset for at least three chimes or a full minute if possible). Then Option Boot the Mac and select the "Install" icon to "resume" the aborted installation process.
Otherwise I highly recommend you just keep using your old OS until Apple resolves these Big Sur installation issues.