You roll/gather up the sock so that there’s just the toe part free and you put that over the toes. Then you very gradually unroll the sock from the toes end while drawing it further onto the foot. You come to the other end of the sock and pull it up neatly.
The idea is not to cause any avoidable friction on the sock-wearer’s foot, and not to leave any little folds or ridges under the foot that could hurt the sock-wearer as they walked.
What do you think of my description of how best to put on a sock for someone else? – Does it seem clear to you?
I think that carers or anybody else who puts socks on for someone else who can’t do it for themselves (too frail, too ill, too fat, painful hands, etc) should be instructed in the best way of doing it to cause least pain or discomfort for the person being cared for.