People with fragile egos are people that have a lot of pride and get offended very easily.
Tending to is synonymous with looking after, or serving.
So, when they have to tend to their fragile egos it means they need to try and look after the authors, meeting their, potentially silly, demands without upsetting them - which is very easy to do because of their fragile egos.
They call it babysitting to imply it can be like looking after (figuratively) the needs of a bratty child. "I want this". "Get me that". "I want some M&M's, but you have to remove all of the red ones because they ruin my chacra". That sort of thing.