The reason for this requirement stems from the master equation:

\$\Gamma_{\sigma}'(t) = H(\Gamma_{\sigma}(t))\$

Since \$$\Gamma(t)\$$ must always remain in the simplex \$$\Delta\$$ -- i.e. must always be a probability distribution -- a big constraint is placed on \$$\Gamma'(t)\$$.

Specifically, \$$\Gamma'(t)\$$ as tangent vector must contain no normal component with respect to the "plane of the simplex." More accurately put, it must be perpendicular to any vector which is normal to the simplex. The simplex vector which is normal to the simplex is the function which maps every definite state \$$D\$$ to 1.

And to say that a tangent vector is normal to this is to say that the sum of the components of this tangent vector is 0.

Now the jth column of the matrix for H is the tangent vector \$$\Gamma_{j}'(0)\$$.

Hence each column must sum to 0.