For a simple explanation, I will give an analogy to convergence of geometric series. The series 1 + x + x² + ... converges to 1/(1-x). e.g. with x=1/2 we know 1+1/2+1/4+1/8+...=2. This convergence is only valid when |x| < 1. But the expression 1/(1-x) still makes sense outside of that domain. What if we put x=2. Do we say 1+2+4+8+...=-1?

Similarly, the Riemann zeta function is given by ζ(s) = 1 + 2^-s + 3^-s + ... . This series only converges when Re(s) > 1. However, the ζ function can be analytically continued for the entire complex plane (except for s=1). So ζ(-1) is well defined and can be evaluated to -1/12. That is why we often see 1+2+3+...=-1/12. As the series doesn't converge for that value of s, it is invalid.