First, let me correct you definition of $\widehat{LT}\!\!_3$: it is the class of depth-3 polynomial size threshold circuits with polynomially bounded weights.
Polynomial size threshold circuits can be converted to polynomial size Boolean circuits. To show this, it suffices to show that any single threshold gate can be computed by a polynomial size Boolean circuit. The proof goes along the following lines:
- Every threshold gate is equivalent to a threshold gates whose weights have bit-length $O(m\log m)$, where $m$ is the number of inputs. This can be shown using linear programming (see below).
- Adding $m$ numbers of bit-length $O(m\log m)$ can be accomplished using a Boolean circuit of size polynomial in $m$.
In our case $m$ is at most the number of gates in the original threshold circuit, which is polynomial (in $n$, the number of input bits). Therefore the Boolean circuit equivalent to any single threshold gate has polynomial size.
The foregoing shows that $\widehat{LT}\!\!_3 \subseteq \mathrm{P/poly}$. While it is also true that $\widehat{LT}\!\!_3 \subseteq \mathrm{NP/poly}$, it seems like a typo in the paper.
Finally, let us indicate how to show that every threshold gate on $m$ inputs is equivalent to one in which the weights have bit-length $O(m\log m)$. Consider the following linear program. The variables are $c_1,\ldots,c_m,\theta$. For each $x_1,\ldots,x_m \in \{0,1\}^m$, if the threshold gate outputs 1 on this input, we add the constraint
$$ \sum_{i=1}^m c_i x_i \geq \theta + 1, $$
and if it outputs 0 then we add the constraint
$$ \sum_{i=1}^m c_i x_i \leq \theta - 1. $$
It is not too hard to check that this LP is feasible, essentially using the parameters of the threshold gate (possibly scaled). On the other hand, LP theory tells us that there exists a basic feasible solution, which is obtained by choosing $m+1$ linearly independent inequalities and treating them as equations. Cramer's rule, which expresses the solution as a ratio of determinants, completes the proof (exercise).
