Roots of the XXZ Bethe Ansatz equation The XXZ spin chain Bethe Ansatz equations are a complicated system of rational function equation:
\[  \left(\frac{\lambda_j + i/2}{\lambda_j - i/2}  \right)^N = \prod_{l=1, l \neq j}^M \frac{\lambda_j - \lambda_l + i}{\lambda_j - \lambda_l - i}  \]
Generically, since there are M equations and M unknowns generically we could say the solutions in $\lambda$ are a discrete set of points.  
Is there any other logic to their roots besides the fact that they solve these equations ?

Spin chains have to do with the raising and lowering operators in the 2-dimensional representation of $SU(2)$ :
\[\left.\begin{array}{cccc} S^+ : & |\uparrow  \rangle & \mapsto  & 0 \\
& |\downarrow  \rangle & \mapsto  & |\uparrow  \rangle \\ \hline 
S^- : & |\uparrow  \rangle & \mapsto  & |\downarrow  \rangle \\
& |\downarrow  \rangle & \mapsto  & 0   \\ \hline
 \\ 
S^z : & |\uparrow  \rangle & \mapsto  & \tfrac{1}{2}|\uparrow \rangle \\
& |\downarrow  \rangle & \mapsto  & \tfrac{1}{2}|\downarrow  \rangle \end{array}\right.
\]
The Hamiltonian acts on a tensor product of $SU(2)$ representations $V^{\otimes M}$.
\[ \mathcal{H} = - \frac{1}{2} \sum_{n=1}^M \bigg[ \sigma_n^x \sigma_{n+1}^x 
+ \sigma_n^y \sigma_{n+1}^y  + \Delta \sigma_n^z \sigma_{n+1}^z\bigg] \]
Various values of $\Delta$ have different interpretations.  Since $[\mathcal{H}, S^z] = 0$ we can diagonalize in terms of the $\uparrow, \downarrow$ states.  The Bethe Ansatz equations are the eigenvalue equations for $\mathcal{H}$ in this basis.
None of this tells you how to solve the eigenvalue equations, here they are.
 A: The Bethe equations and Bethe ansatz are the central tools to study quantum integrable systems and  there are thousands papers devoted to them. I cannot pretend to know substantial part of this research, so let me write some remarks which I am aware of.
You ask  "... is there logic to their roots ... "
Yes, there is some logic - the keyword is "string hypothesis" (it is NOT related to string theory).
Let me quote from String hypothesis for gl(n|m) spin chains: a particle/hole democracy
Section 3.1 page 10.

Suppose that N is large and some Bethe root $\lambda_n$ has a positive
imaginary part. Then the l.h.s of (26) is exponentially
large with N. To achieve this large value on the r.h.s. there should
be another Bethe root $\lambda_n′ ∼ \lambda_n − i$, with the help of which the pole in
the r.h.s. is created. Repeating the same arguments for $\lambda_n′$ and using
the reality of solution of the Bethe Ansatz [36], we conclude that the
Bethe roots are organized in the complexes of the type:

$\lambda_k = \lambda_0 + ik, ~~ k= -(s-1)/2, -(s-3)/2, ..., +(s-1)/2. $

where s is an integer. These complexes are called s-strings.
String
hypothesis in its strong form states that all solutions of the Bethe
Ansatz equations can be represented as a collection of strings, and
that $\lambda_k$ are approximated by $\lambda_0 + ik$ values with exponential in N
precision. In its strong form the string hypothesis is wrong. However
there is an evidence that its weaker version is correct if the proper
thermodynamic limit is taken. The weaker version states that most of
the Bethe roots are organized into strings with exponential in N
precision, and that the fraction of solutions which signiﬁcantly diﬀer
from (27) decreases to 0 when N → ∞. We discuss in more details
applicability of the string hypothesis in appendix A.

