You don't need to.  Filter paper is only folded if you need to fit a flat two dimensional piece of filter paper into a 3-D conical funnel.

Increases the amount of surface area available for filtration.

Increases the amount of surface area available for filtration.

So, could you tell me that what's the surface area difference between 8-folded and 16-folded filter papers? (or even an unfolded one)

Of course I can. There is no difference in surface area.

However, there is definetly a larger surface area than if you were to simply fold the paper twice and place it flat against the sides of the funnel, do you agree?

As I learned that in secondary school and demonstrators in my university, one of the advantage of increasing the folds of a filter paper will give a higher efficiency of diffusion by increasing the surface area.

I wonder know how can it increase the surface area?
Their surface areas should be the same I think.

Assuming "R" is the radius of the filter paper, no matter the no. of folds it is, the length of the folded paper is also "R".
OK, if there are 2 filter papers (+ 1 funnel each), one is 8-folded, another is 16-folded, when the mixture is poured into them as same level, for example 1/2R, their areas are the same (assume there is no overlapping between each folding section), am I right?

One thing I want to point out is that, as the folds of filter paper increase, the volume of the cone (just imagine it looks like a cone) decrease, do you agree?
You can see that the cone is smaller and the angle of it is also smaller than less folded one.

For same area but difference in volume, therefore, the surface area to volume ratio increases as no. of folds of filter paper increases, just like the case of diffusion of red blood cell that why it is biconcave in shape. This give a higher efficency of diffusion (filtration).
Do you agree?

I wonder know how can it increase the surface area?
Their surface areas should be the same I think.

If you put filter into the funnel without 'fluting' the surface will be that of a funnel cone. If you 'flute' the filter it no longer touches funnel with it whole surface, and the surface used for filtration is that of the whole filter - thus larger. No matter how many folds are there, the surface will not change of course. That's the same situation as with the air filter in the car, it is often made of folded paper.

What is not clear to me is if the larger surface will be more effective when it comes to speed of filtration, as with folded filter you are loosing sucking from the funnel leg. Probably it all depends on the type of precipitate you are trying to separate.

I am really interested now as to what prompted the question. What were you trying to filter?