Problem with text quality in finished video in After effects

I have no idea what I am looking at here. I assume that is a portion of a screenshot at 100% so I am seeing the actual pixels. If that is the case there are some obvious problems with your font choice and color.

I took your image into AE and magnified it in the comp panel so I could see the actual pixels and make some notes and observations. I also added a little color correction so it is easier to see the problems in the rendered footage.

Here's how video works. Video is pixels, text is vectors turned into pixels. If pixels don't line up with pixels they get interpreted or averaged so lines that are 1 pixel thick must be perfectly lined up with the pixel grid or they will change color and luminosity. A 1-pixel thick black line centered halfway between two pixels will be interpreted as a 2-pixel thick gray line.  If the line is moving it will flicker between gray and black. Your font is a very bad choice for video because it contains a lot of very thin parts that will be extremely difficult to line up with the pixel grid.

Now let's talk about color. MPEG compression used in the standard h.264 MP4 files you distribute does not contain the original pixels from the comp in all of the frames and the color values for those pixels are always averaged. There are I-frames with all of the luminance values for every pixel in that frame, but the color values for those pixels are sampled from at the very best settings blocks of 4 pixels. The other frames are called P and B frames and they are predicted by algorithms that look at the differences between the I frames and guess at where the pixels should be and what color they should be. This reduces the file size by at least 3 times but causes motion and color artifacts in the image.

When you put a 100% red text layer against any other background there is no room left in the red channel so the adjacent pixels have more influence from red than they would if your red text had a value of 235 or even 200. That is where the red glows come from that I circled in the screenshot. Just reducing the red value in the color will go a long way to minimizing the color artifacts when the comp is rendered. 

If you want a small font you should be using a sans-serif font. If you must have a sans-serif font either add a small stroke to the font to thicken it up or choose one that is thicker. In any case, your font must be thick enough that the thinnest parts are at least 3 pixels wide. Even if you render with the highest data rates and use lossless formats, thin lines are going to be problematic. You just have to design around the limitations. 

Multi-pass compression may help the text a bit, but until you fix the thin lines problem and fix the color problem, your text is going to look bad on your phone. There is no way around it. Handbrake is not going to do any better job than the Media Encoder rendering this project. There is no advantage at all.

One more thing. In the future please use the toolbar to embed images so we don't have to download them to see them. 

Tell us composition specs and how you set the Output settings. Looks like you are working on a composition format and setting a bigger Output, if so, obviously you have distortion.