It is true that $\zeta(-1)=-1/12$. But this doesn't imply that
$$
1+2+\cdots=\frac{-1}{12}
$$
For one thing, the l.h.s. is undefined if we stick to the standard meaning of $+$. You could, if you want to, extend the definition of $+$ so that this is equation is true. But this is highly non-trivial: the standard rules of addition break down in this generalised notion of summation.
In QM (and string theory in particular) we sometimes have to consider improper sums, and they sometimes turn out to diverge. We don't like infinities for the obvious reason: a theory with divergent objects is ill-defined.
We therefore introduce a regularisation, which amounts to define a certain (context-dependent) deformation of the basic objects so that everything is finite. You perform some (rigorous) algebraic manipulations in terms of finite objects, calculate something measurable, and in the end you remove the regulator (by changing the deformation into the identity transformation). Sometimes this lengthy process is summarised as
$$
1+2+\cdots\to -\frac{1}{12}
$$
which does not mean that these two objects agree; it means that you can replace your naïvely divergent sum by the r.h.s., and in some cases the result you get is right.
This is clearly non-rigorous if you don't go through all the details of regularisation and renormalisation, but it is a very solid mathematical procedure once you know what it actually means. I am sure that you'll remain skeptic until you calculate something yourself. You shouldn't believe people if they tell you "it can be justified"; you should check it yourself. But in order to do so, you'll have to learn the basics, for example, QFT.
Divergent sums and integrals appear everywhere in QM, but for some reason the popular choice to introduce the weirdness of QM to the general audience is $1+2+\cdots=-1/12$. There are thousands of more involved, more interesting and important examples, but this one here looks funny and so this is the one you'll find everywhere online.
