What is the meaning of "self-conscious"?

Solution 1:

This is definition 2 from Merriam-Webster

uncomfortably conscious of oneself as an object of the observation of others

or definition 1 from Lexico

Feeling undue awareness of oneself, one's appearance, or one's actions.

Your example about your appearance is simply a specific example of this, but there can be many reasons why you can feel self-conscious. In the quoted article, reminding someone that you're aware that they're going through something unfortunate can also do it. This results in a conundrum: you'd like to comfort them, but raising the issue can cause discomfort.