Why don't plant cells burst when placed in pure water?

This is because they have an extra structure that is not in animal cell. They have a cell wall.

When water enters the plant cell due to osmosis, the cell membrane expands and pushes against the cell wall. The cell wall protects the plant cell from bursting. In that way, the plant cell would only be turgid due to the turgor pressure.
In opposite, animal cell doesn't have a cell wall so if extra water flowed into the cell, the cell membrane break and the cell bursts.

Cells of an organism tell each other when predators are near.