Learn Hardcore Japanese: okonomiyaki wo kiriwake, torizara ni oita. - I cut the okonomiyaki into portions and put it on the small serving plates.

切り分ける means to cut something into pieces and separate/portion it out. In this sentence, it suggests not just cutting the okonomiyaki once, but dividing it into servings.

So:

お好み焼きを切った = cut the okonomiyaki
お好み焼きを切り分けた = cut the okonomiyaki into portions

That makes 切り分ける a very natural verb here.

Why is there an を after お好み焼き?

The particle を marks the direct object of the verb.

Here, お好み焼き is the thing being cut up, so it takes を:

お好み焼きを切り分け
= cut up the okonomiyaki

A natural English way to think of it is: What did they cut up?
Answer: the okonomiyaki.

Why is there a に after 取り皿?

The particle に marks the destination or place where something is put.

So:

取り皿に置いた = placed it on / onto the small plate

Even though English often says on the plate, Japanese commonly uses に for the target location of placement.

What does 取り皿 mean?