I have dabbled a bit with plant physiology, perhaps I can provide a simple explanation:
Auxin's are plant growth hormones (one of the most familiar is IAA indole-acetic acid).
Living plant cells are under a high turgor pressure from electrolytes present in their cytoplasm. The auxin IAA, increases the elasticity of the cell walls and because they are under high turgor pressure, they elongate.
One explanation is the so called
Acid-Growth Hypothesis.
Here is how it works. IAA activates a proton (H+) pump in the cell membrane, causing H+ to move into the cell wall, increasing its acidity. (An increase in hydrogen ion concentration decreases the pH and therefore increases acidity). This low pH activates the enzyme
expansin (appropriate name

), which breaks bonds within the cellulose fibers. With these bonds broken, the cell wall becomes more elastic and expands under the turgor pressure.
You may have noticed, when you put a plant near a window, after some time, the leaves point in direction of the bright light. If you then turn the plant around 180 deg, after some time, the leaves bend upright and continue to bend once again towards the bright light. This is a the
phototropic effect. One explanation of this, is that auxin is being released in the cells on the
dark side of the plant and by the "acid growth hypothesis", those cells elongate causing the leaves to bend towards the light.
For a more scientific description, see
ref especially sections 3,4 & 5