Why MIT lecture breadth-first search algorithm is so complicated?
I think this is best handled by contacting the lecturer, but here are four reasons.

A Kain0_0 remarks, your solution is recursive, while his isn't.

Didactic considerations behind this might be:


Recursion is hard for some students. Perhaps it hasn't been taught at this point in the course, or is considered an unnecessary burden.
Most BFS implementations in practice do not use recursion. Programmers need to spot algorithms in other peoples' code all the time. There is value in presenting algorithms in their most common form.


Furthermore,


You're sweeping some stuff under the carpet. He makes his data structures explicit, because he wants to show how much time and space his algorithm uses. You don't: you leave the recursive call stack implicit, and you don't show how .children and .index are represented or created.
He's doing more. He can report the level at which the node was found. You don't keep that information.