There are a number of ways to do this:
You could pass the text through a mask that is made of noise (use luma matte) and then animate the contrast of the noise so it starts as all black and then it eventually "whites out". This creates text with holes in it that fill in.
For more of an edge effect, premcompose a copy of the text, blur that text and animate the blur so that the text blurs so much you can't see it and animates to sharp. Combine that with a noise in the precomp so you get a ratty edge- you may need to scale the noise to keep up with the blur of the text. Then change the contrast with an adjustment layer to have the precomp's white areas expand or contract and use the precomp as the mask that bring on the text in the main comp. Then you can fade out the noise or use the contrast in the noise trick above to make the noise fade with the animation. This will bring the text on from the centre of each character out, like a water color bleed.