What you're seeing is a product of the very bad color resolution in H.264. It becomes most apparent when there is one solid color against another, like your white text against pink.
You may not be able to get rid of it. It depends on what you are required to deliver out of After Effects, so please tell us. But understand that you may be stuck with it.
Dave already said it - you are seeing an inherent limitation of MPEG-based compression algorithms (and many others' fort hat matter) - color quantization. You can't avoid it entirely, but you may be able by choosing more suitable colors.
You can also help disguise it by adding some texture to your images. Whether its over the whole thing or only over certain elements, adding texture really helps break up issues like this. Bonus, it's also more visually interesting.