What was the eye opening experience that Katniss had years back when she eyed a dandelion at the school yard?

The eye opening experience that Katniss has, after seeing the Dandelion, is that there is still hope, and that her family has a future, even though her father has died, her mother is in a deep depression and she has a younger sister to feed. Bear in mind, that her family was starving to death (literally) her father (the source of money in the family) had just died, her mother was deeply depressed and unable to be a mother, so Katniss had to become the responsible adult in the family, even when she was only a teenager.

Specifically, the Dandelion makes Katniss realize that she can go to the woods to collect edible plants (like dandelions) and then she starts hunting and she learns how to use a bow expertly, skill that will saver her life in the hunger games.

Later on, Peeta becomes the symbol of the dandelion. (Spoiler alert) She ends up saying that what she needs in her life is "the dandelion on the spring, the bright yellow that means rebirth instead of destruction, the promise that life can go on no matter how bad our losses".