You can use an old film flash for the background, but honestly it's mostly just a clever distorted shape/solid layer and some blurring. The grain helps, but it's so quick that it might not matter. First, I'd suggest downloading the video you're referencing and going frame by frame through it. That's how I learned how to do a lot of things back in the day. I learned a lot about how I can cut in things quickly without it being apparent to most people. And you can stop to take a look at it to see what's happening.
Here's a quick set of instructions, but play with it.
1. Create a comp.
2. Put your logo in.
3. Make a new shape just smaller than the comp (so that you can blur it and it doesn't get cut off).
4. Set the fill of the shape to a vintage looking color.
5. Set a key.
6. Right click on the key and make it a hold keyframe.
7. Change the color once or twice.
8. Key a fill effect on the logo in the same way (at one point they might've inverted the colors as well).
9. Add an adjustment layer with fast blur and key the blur amount the same way.
10. Bring that comp into your main comp. Put it where it needs to go and resize it.
11. Add the corner pin effect.
12. Set keys for the corner pin just before the logo is "locked up."
13. At the in point of the layer, spread the top two corners of the corner pin out in your comp.
14. Select those keys and right click to make them all hold keyframes as well.
15. Play it back a few times and adjust to taste.