As light intensity increases, what happens to the rate of photosynthesis?
Photosynthesis, specifically the light reactions, require light to occur. The light's photons excite the electrons in the pigments of the photosystems which activates the light reactions portion of photosynthesis.
The more light there is, the more photosystems in the thylakoid membrane can be activated.
However, light intensity can only increase up to a certain point before the rate of photosynthesis no longer increases.
Once there is a sufficient intensity of light, the ATP and NADPH that come from the light reactions will be in abundance. For the remaining part of photosynthesis to occur (the Calvin cycle), carbon dioxide is needed.
Even if more and more ATP and NADPH are being formed, they will not be able to act if more carbon dioxide isn't entering the plant.
Thus, the plant reaches a "light saturation point" and the rate of photosynthesis is limited due to a limited amount of carbon dioxide, or due to some other limiting factor.
The following image provides more nuance. It shows that light intensity and the rate of photosynthesis increase with one another. It also shows that the rate at which photosynthesis levels out is dependent upon other factors—both plants in 0.1% CO₂, they cannot photosynthesize at nearly the rate of the plants in 0.4% CO₂.
Similarly, plants photosynthesize at a greater rate in higher temperatures (generally—not in temperatures that are too hot—this is also dependent upon species).