TL;DR at bottom

Plants "respond" to stimuli through a process known as tropism, and two or possibly three of these stimuli are present: thigmotropism (response to movement or touch), phototropism (response to light), and chemotropism (response to chemical stimuli; I list this one as UV light may breakdown substances on the surface of the plant to trigger a response).

These responses aren't so much a thought process as much as a reflex, such as if your knee reflex triggers without thought if hit in the right spot. Plants responding to these stimuli have what I'd describe as spring-loaded traps on the surface of each cell, ready to trigger with the right application. It is much like cell receptors waiting for a specific hormone to bind to a site before expressing a response.

When that trigger is met, a group of ions makes it way down a pathway to make a cell to increase it's uptake of water or decrease its uptake (this is through osmosis, and creating a hypotonic or hypertonic environment in the cell). When you have groups of cells in one region suddenly increase their water uptake and others decrease the amount they hold, the plant will flex in a certain direction, resulting in plant movement. These movements usually aren't drastic as with specialized plants like touch-me-nots and Venus flytraps. With the pea plant it would occur over hours.

Interesting to note that touch response can also influence the final structure of a plant, where it will alter how it grows , known as thigmomorphogenesis. These growth response are exaggerated at the tips of roots and tendrils, or meristems. This is how you will find trees that have branches arranged around structures or influenced by wind.

TL:DR: plants have specialized receptors for light, touch, and chemicals that tell groups of cells to increase or decrease water uptake making the plant bend.