Why does solving the underlying polynomial system "break" the multivariate cryptosystem
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.