Growing out hair or nails takes time, so the phrase implies waiting for natural growth. She's growing out her bangs, so she's using hair clips meanwhile.
Pro Tip 2/3
Transition Phase
"Grow out" often refers to the awkward phase between two lengths or styles. He's in that awkward stage of growing out his buzz cut.
Pro Tip 3/3
Personal Choice
The decision to grow out something is usually a personal or aesthetic choice. I'm growing out my nails for a new look.