What is the proof that boolean circuit (no negation gate) can be arranged as alternating OR and AND gates
Let $r$ be the root of your circuit (without loss of generality, an OR gate), and let $V_i$ consist of those gates at distance $i$ from $r$. Assign (in your mind) a layer for each gate: layer 0 for the root, layer $2i-1$ for an AND gate in $V_i$, and layer $2i$ for an OR gate in $V_i$. The result is almost what you want – the only difference is that edges could span more than one layer.

In order to fix the problem, for each gate $g$, create gates in all layers above it (i.e., with smaller number) of the correct type, and connect them in a path (so all the new gates will have in-degree 1). Now you can arrange that each edge span exactly one layer.

Starting with a circuit of depth $D$ and $n$ gates, we have created a circuit of depth $2D$ and at most $nD$ gates. In particular, if the original circuit had polynomial size then so does the new one (with a different polynomial), and if the original circuit had constant depth (or logarithmic, or polylogarithmic) then so does the new one.