Remove black flickering in After Effects

There is no one click solution. The best you can hope for is to create a procedural matte, a matte created using a bunch of effects, and then use that matte to limit color correction to the areas you want to change. It's the only solution that I know of that doesn't require a bunch of roto which would be extremely time consuming on this shot.

To give this a try I took a screenshot of a frame of the video, created a comp from the screenshot and then used curves to generate a high contrast matte layer to be used as a track matte for a black solid. These layers were pre-composed so I could use some matte tools to refine the matte and an adjustment layer with nothing more than Hue/Saturation - Colorize added above the original footage.

The Pre-comp was used as an alpha track matte and Matte Choker was used to adjust the edges of the matte and clean them up a bit. A little tweaking and less than 5 minutes and you have some selective color correction applied to your footage. This is far from perfect and can be refined by additional techniques but it should give you a place to start. Because of the fine edges you are working with I'd convert the project to 32 bit to give you more control over the matte edges. The quality of the procedural matte is going to determine the quality of the color correction. That's the most important part. There are a bunch of ways to achieve this, curves was just the first tool I tried to see if it were possible.

Here's a screenshot showing the modified properties of the layers in the pre-comp and the main comp to get you started:

I hope this helps.