OK, I found the original thread and read through it:
http://www.whatsbestforum.com/showthread.php?16365-The-sum-of-all-natural-numbers-1-12
Looks like the ultimate conclusion was not reached. Here is my read of it.
It is useful to be able to represent infinite sums as some value as to allow further analysis of physical world around us. The proof used to drive these answers must be correct at some level as the physics based on it works.
In some sense then, this sum may be a technique to solve a problem using much simpler math. Indeed there are much more complicated ways of proving some of these equations but why not use this simple concept to arrive at the same place quicker and faster?
Another similar concept is that of complex numbers. i = square root of -1. Such a concept is absurd as you can't take square root of -1. But we can use that to solve countless problems.
And remarkably just like the sum of natural integers being -1/12, we can use square root of -1 to solve real physical problems where this very value, actually means something in reality. In electronics, imaginary part of an equation represents the phase of an AC signal. In other words, it is not imaginary at all! By being able to encapsulate phase in the same number as its amplitude, we are to represent the entirety of an AC waveform and manipulate it using math of complex numbers.
The same scheme is used in string theory to come up with a sum of ways a string can resonate which is the original frequency and infinite multiples of that frequency. That sum instead of being infinite, can be considered to be 12. Multiple that by 2 and you get 24 for each dimension. Add the original X and Y plane and the total number of dimensions needed to express this infinite variety just becomes 26!
There: you have my plagiarized and embellished explanation of why this is a valid sum and so useful in real life.