Apologizing to your child when you have reacted harshly to them is very important, for your child, and for you. You are providing a powerful model for taking responsibility for one's actions. It also means that you are willing to step back, reflect and work on your own self-regulation--one of the most necessary and greatest assets that will strengthen your loving and secure attachment to your child. 

But, in my practice, I often see parents apologizing to their children when they have done nothing wrong. A child accuses his mom of playing the game "wrong", meaning he did not like the outcome and lost. Or, dad wasn't first in line at carpool pickup. Or, mom made the pancake the "wrong" way; the shape isn't right. It's not a perfect circle.

In these moments, children are distressed because something doesn't happen the way they expect or want. They have very fixed ideas about the way things should be, and when their expectations are not met, they fall apart. One two-year-old flipped out when her mother wore her hair up. She would insist her mom take it down, which her mom acquiesced to, to relieve the stress.