To "feel self-conscious" means to worry a lot about what other people think of you. In the example above, if the speaker had laughed at her daughter, her daughter would have felt self-conscious about her mistake.

As another example, imagine that you're at a party where you don't know many people, and you're dressed up a lot more formally than the other people there. You can describe your feeling as:

I felt extremely self-conscious the whole night.