Why does solving the underlying polynomial system "break" the multivariate cryptosystem I was wondering why exactly does solving a polynomial system (directly or indirectly) "break" a multivariate cryptosystem as a digital signature. 

I realize that the exact reason differs from system to system, but in general, from what I can see, it then allows attackers to sign messages with a private key that is not theirs. Am I on the right track here?
Multivariate cryptographic schemes that perform digital signatures like HFEv, FLASH and Quartz have something in common. As opposed to enciphering data like in a normal cryptosystem where a public multivariate polynomial $P(X)$ is given as:
$$P(X) = P(p_1(x_1,\ldots,x_n),\ldots,p_n(x_1,\ldots,x_n))$$
You input the plaintext bits $x=(x_1,\ldots,x_n)$ into $P(X)$ right? Well when dealing with digital signatures we do the opposite. This is, as we are the owners of the construction, given $Y=H(m || salt)$ where $m$ is a message, we can find the plaintext tuple that sends $X$ to $Y=H(m || salt)$ by inverting the construction:
$$X=P(Y)^{-1} = S^{-1} \circ \varphi^{-1} \circ F^{-1} \circ \varphi \circ T^{-1}(Y)$$
which clearly yields $X$ such that $P(X)=Y$. Now Alice sends the tuple $(m,salt, Y,X)$ and Bob verifies as $P(X)=Y=H(m||salt)$.
Thus if Eve wants to forge digital signatures to trick Bob into thinking she's Alice, Eve must solve either the $\mathcal{MQ}$-Problem or the Isomorphism of Polynomials ($\mathcal{IP}$) which both are reasonably hard.