There have been a lot of studies done on allelochemicals. These are chemicals that plants produce that can suppress the growth of other plants. The general consensus seems to be that healthy, fast growing plants produce enough allelochemicals to inhibit algae growth. On the other hand, algae is well adapted to growing in low light and nutrient poor conditions, so if you let it get established and your plants are weak, the algae itself can produce allelochemicals that inhibit the growth of plants, killing them. Not only that, algae on the leaves blocks light starving the plants.
So, adding Co2 simply removes the carbon limiting growth factor of plants, allowing them to grow healthy and fast enough to produce enough allelochemicals to inhibit algae growth. It's like chemical warfare between plant species.
Note: this also explains why some types of plants just won't grow in a certain tank. The plants already established might be producing allelochemicals that wipe them out, or maybe the water chemistry is not favorable for a certain type of plant.
“We don't stop playing because we grow old; we grow old because we stop playing.” - George Bernard Shaw