condensed matter - Spontaneous Symmetry Breaking Field - Physical Significance.

One way to define spontaneous symmetry breaking (SSB) is as follows (Morandi, Ercolessi and Napoli, 2001; my wording):

We can define SSB as occurring when: $$\lim_{h\rightarrow 0} \lim_{N\rightarrow \infty} \langle\mathcal{O}\rangle_{N,b}\ne 0\tag{1}$$ where $\mathcal{O}$ is some quantity which is not invariant under the symmetry.

I understand mathematically why this order of limits must be taken (rather then having the $h$ limit first). But why is this the case physically and why does the condition (1) correspond to SSB?