Basically what happends is that the pikachu emits the smoke created by running, so you do some sort of technique that makes you run in place, so smoke is created all around you. I always found this really cute yet cool at the same time, anyone have any idea how to do this? I realize pikachu isn't the only one who can actually do this, but he is the first one I saw do it.
This technique is called dash dancing, and can be performed with any character by rapidly tapping the control stick left and right.
From the Super Smash Bros Wiki page:
Performed by rapidly tapping the analog stick left and right while on the ground, it cancels out the character's initial dashing animation with another animation in the opposite direction, causing the character to quickly and repeatedly dash to the right and left in a short distance.
The time window in which a character can dash and then change directions by dash-dancing matches the amount of frames in their initial dash animation. After they complete their initial dashing animation, they'll transition to their standard dashing animation; at this point, tapping the analog stick in the opposite direction will cause the character to enter their turnaround animation, during which no attacks can be performed.
In Smash 4, performing a dash dance results in the emission of smoke effects from the dash animation, as you noted, will stack up to make a pretty cool effect.
Apart from being just a cool trick to do, dash dancing was effective in Melee to play spacing mindgames with your opponent. However, due to various changes over the series, dash dancing is of little competitive use in Smash 4.