I'm about 90% sure that in this case it means the equivalence class to which the likelihood function belongs, where two functions are equivalent precisely if either is a positive scalar multiple of the other. Notice that
$$
\Pr(Y_1=y_1\ \&\ \cdots\ \&\ Y_J=y_j) = \frac{(y_1+\cdots+y_J)!}{y_1!\cdots y_J!}\cdot \prod_{j=1}^J \left(\frac{e^{\beta_j}}{e^{\beta_1}+\cdots+ e^{\beta_J}}\right)^{y_j}
$$
As a function of $\beta_1,\ldots,\beta_J$, this is the likelihood function
$$
L(\beta_1,\ldots,\beta_J) = \text{constant}\cdot \prod_{j=1}^J \left(\frac{e^{\beta_j}}{e^{\beta_1}+\cdots+ e^{\beta_J}}\right)^{y_j}
$$
where "constant" in this case means not depending on $\beta_1,\ldots,\beta_J$. For many purposes, the values of such "constants" don't matter. The celebrated "likelihood principle" is a more-or-less philosophical position saying all statistical inferences should be based only on what I am surmising this author means by the "likelihood kernel". Certainly the function he's written is the simplest representative of the right equivalence class, which is what any sensible person would use.

Two contexts in which the statement that the value of that constant doesn't matter can be made into a precise mathematical statement and demonstrated to be true are (1) maximum-likelihood estimation and (2) finding a posterior probability distribution (in this case a distribution of $\beta_1,\ldots,\beta_J$) by multiplying the prior probability distribution by the likelihood function and then normalizing (Bayes' theorem).