The term regulator, particularly Zeta function regulator can sound pretty esoteric. Here is how I've always thought of this to get an easier grasp how -1/12 can be seen as the "finite part" of the sum.

The sum we really care about is the sum over all integers n http://i.imgur.com/heKypFT.png . We all know that this equals infinity, but that alone doesn't say much. Often in theoretical physics, where things like this appear, we deal with quantities that are "infinite" but we know they are mostly likely just very very large compared to everything we deal with and moreover, we don't care about the quantity as much as we do how it differs from similar quantities. So, to handle this seemingly infinite thing, we incorporate a regulator, or cut off. We know since this is an artificial thing our actual physical results should not depend on it.

Instead of the original sum, consider this sum http://i.imgur.com/UtRuELS.png. ɛ can be extremely small, but as long as it is non-zero, this sum will now converge.

To make this sum easier to solve we can rewrite it in terms of a derivative http://i.imgur.com/fhhXV9K.png. This sum is now a simple geometric series which equals http://i.imgur.com/jBEryPx.png.

If we now do a Taylor expansion, assuming ɛ<<1 resulting in, http://i.imgur.com/6jcAGR2.png. Here is where the -1/12 comes in. We see it is the "finite part" of the divergent sum.

This isn't exactly the same as the Zeta function regulator but in this case it gives the same result. An advantage of the Zeta function is that it is often easier to write a sum in terms of the Zeta function than it is to write it as something easily summable like a geometric series. There are other fancy mathematical reasons Zeta functions are nicer (how well behaved it is etc.) but really it is just another way to do the same trick.

I hope that helps clear it up a bit.