transfer files from external backup; time machine

I backed up my old macbook air (2014, big sur 11.5) to an external drive via time machine. I want those files on my macbook pro (2020, big sur 11.6). How do I move them from the external drive to my newer macbook? if not move them, at least access them?

In the alternative, can I backup my macbook air to icloud and grab the files I need that way.

When I set up my macbook pro, it did the put your computer next to it for setup thing. Did that get my files? I see some but don't want to wipe my macbook air for trade it in without being sure.

Thanks.