Doubletime feel means that you double the speed of your accompaniment but the time length of the bars (and thus the speed of the melody) stays the same. That is, for a bar in double time you sort of play your accompaniment of a non-doubletime bar twice at double the speed.
Here's an example to illustrate the idea, with the same passage played as regular time and as doubletime:
Note that the tempo (BPM), the melody, and the chord shifts stay the same while the rhytm section gets busier.
You don't necessarily have to double everything in the accompaniment (as is done in the example); you can just play in a way that it feels like the beat doubled.
For the song you linked you notice that the speed that you want to count 1 2 3 4 doubles where the distortion enters, but the speed of the melody doesn't double! You could think of this as if you start to count eighths instead of quarters. That is for the length of one notated bar you will count 1 2 3 4 5 6 7 8.
Does it have any impact on the tab (tempo, time signature, rhythm feel) or playing?
It seems like your pdf file has notated everything for you so you can consider the "doubletime feel" to be an explanation to what is happenning - why the accompaniment suddenly seems busier.
The tempo of the quarter notes stays the same!
The time signature stays the same!
The tempo of the vocals/melody stays the same!
The only difference is that the guitars, bass, and drums simply play more notes/beats per bar in a fashion such that it feels like the tempo doubled!