In your picture, that may have been achieved in 2 ways. One, it looks to be a particle emitter(or multiple emmitters), all it would take is setting the initial color and then having it change to white over time.
Another method would be to place a adjustment layer over the video, with a black and white color effect applied. Then mask the adjustment layer halfway across the screen and play with the horizontal feather property.
And if there is anyway possible, could someone teach me how the color can follow the moving object and make the whole screen in colors while the moving object moves in/across the scene?
Here's one way you could do it(i'm sure there are many others) Let's use a car moving across the screen. You would drop your video into a composition, and duplicate it. On the bottom layer, you would add the Black&White effect. The top layer, you would need to mask/rotoscope the car as it moves across the screen, and use that as wipe revealing the color layer. This type of effect works best if you have an object that that stretches from top to bottom if it's moving horizontally, left to right if moving vertically. Then you just have to mask the front edge as it moves across the screen, then just having the mask eventually fill the entire screen.
"I have not failed 700 times. I have succeeded in proving that those 700 ways will not work. When I have eliminated the ways that will not work, I will find the way that will work."
---THOMAS EDISON on inventing the light bulb.