Auxin's main function is to stimulate cell elongation and does this through ABP1. ABP1 is responsible for the membrane transport processes (such as transporting H+ across the membrane) and if Auxin boosts ABP1, then the membrane can allow more nutrients to pass through the cell which eventually leads to the cell elongation. Interestingly enough, too much auxin inhibits cell elongation as it inhibits cytokinins.
The same goes for roots as well. Auxin, in small amounts, will increase the initiation of roots. In large amounts, it enhances adventitious root formations which will often lead to positive effects such as sturdier support in larger trees.
Auxin helps in phototropism. Sides that are unable to absorb light produce auxin to help stimulate the growth so the plant can absorb as much sunlight as possible.