Basically algae spores are present everywhere, and in the presence of adequate light, they are able to extract nutrients (nitrate being a biggie) from the water very efficiently, and that is why when you upgrade your lighting you will almost always get new algae growth - sometimes taking over the tank.
Plants use the same nutrients, but lacking other micro and macronutrients they are less efficient at getting at it, and they get covered in algae too. You need to provide these nutrients to the plants so they become more efficient at getting them from the water, and the algae will starve. There are many factors at play, but the primary nutrient that plants need is CO2
. In high light they can't photosynthesize properly without CO2
. More plants are not necessarily the answer, but providing more nutrients to the plants you have are necessary. Sometimes if you don't have particularly high light but have algae, getting some fast growing easy plants will solve an algae problem, since they grow so fast they suck all the nutrients up instead of the algae. They will often be listed with online retailers as "algae buster" plants.
Hope that was not too much info, and I hope you aren't sorry you asked...