The answer comes down to a balance of energy. Einstein showed us that photons have mass, and many photons of light (and other wavelengths) do hit the earth every day. It is easier, though, to think of the energy that is hitting the earth every day, and ask the equivalent question, "Why isn't the earth getting more energetic (hotter)?" As you know, this is the same as asking why it isn't getting heavier.
The reason is that the earth emits energy itself back into the universe - if it has too much, it "sends it back", so to speak. Generally, the light from the sun warms the surface of the planet (land and water included), and objects that are warm tend to give off (or "radiate") energy - the hotter they are, the more energy they give off. So the whole planet is emitting energy, and some of this escapes back into space (some of it doesn't - it sticks around and heats the surface of the earth by several degrees, making life possible; this is what they call the "Greenhouse Effect"). This isn't the only source of heat in the earth, however - most of our "warmth" comes from internal heating deep in the earth's core due to the radioactive decay of things like uranium. Now, the earth has been around long enough (4.5 billion years), that it's reached a point where its energy "loss" is equal to its energy "gain" - a situation technically called "Thermodynamic Equilibrium".
This is much like the inside of your kitchen oven after you've had the heat on for a long time and the door shut. So, even though a tiny amount of mass ("tiny" compared to the mass of the earth) is added to the planet every day, an equal amount escapes the planet, and we don't get any heavier, or more energetic (unless the Greenhouse Effect is actually increasing due to manmade chemicals in the atmosphere, but that's another story).