If you do a primary color correction on your video to get it normalized (this is assuming your footage looks correct in terms of exposure, white balance, etc); I'd add the following filters to achieve that look without any plugins...
You would have to play with the parameters; I haven't seen your footage, so I can't say how strong a filter would need to be applied.
Either ADD and Adjustment Layer on Top of your Footage Layers; or you can nest your comp into another comp and do your color correction there.
USE THIS ORDER ON YOUR FOOTAGE/ADJUSTMENT LAYER GOING TOP TO BOTTOM
Color Correction - Brightness Contrast: Reduce Contrast (you will have to play with this as I don't know if your footage is High Key to begin with). But to get that look, you want low contrast footage behind your text, so that you lift your blacks and shadows up. I wouldn't increase Brightness unless you shot 4:4:4. But you want very low contrast which basically flattens out the overall image and mutes the colors as well.
Color Correction - Hue Saturation Lightness (HSL): Reduce Saturation by 50%. Increase Lightness: 2-10% Leave Hue at 0.
Color Correction - Vibrance: Now that you've reduced your Saturation; adjust the Vibrance Up till you get the colors to punch through to your liking. You may need to reduce Saturation in previous filter or not reduce it so much as 50, depending on your footage. But pulling back your saturation a decent amount, and replacing it with Vibrance will give you the tonality you're looking for color space wise.
Last Filter: Add Video Limiter Filter, and soft knee your Luminance to 99.90.
Lastly; if you want to knock your TYPE out from your footage background, you can add even a very small amount of Blur. Gaussian Blur; or Fast Blur, Lens Blur is nice but adds render time. All options would work for this.
FOR YOUR TEXT
Adding a even small stroke will help it pop. Use video/film friendly fonts, no fine lines/hairline fonts. Some of my favorite fonts for video are: Gotham Ultra; Gotham Book, Gotham Medium, Whitney (SemiBold, Book, Book SmallCaps) Operator in Small Caps various weights, and if you want Serif, Mercury is a great font, as is Perpetua with the right weight.
If you are using fine line fonts, such as Gotham ExtraLight, adding a stroke of 1px as the same color of your font, and reducing the opacity of the stroke to 50% will help.
And lastly; adding a drop shadow (very subtle so almost not noticable) with 0 Distance, 0 Spread and Size 10-25 (Depending on Fontsize) and then reducing the opacity of the shadow to 10-40% works well. I particularly don't like the drop-shadow look; but I do use it to get the text to pop out from my motion backgrounds.
Regardless, you want your text to background footage to have a wide latitude of contrast. So if your footage is bright, you'll either want to darken the footage down, or use a solid color layer on it, something to create a wider spectrum if you plan to use a light colored or white font. That-- or use a dark font.
If your footage is dark, either brighten it up, or use a bright colored font. Same goes for color.
I would recommend having your project comp set to 32 Bit Floating Point as far as workspace, even if your footage is 4:2:0. You'll get better CC results.
Hope this helps.