The teacher here redrew the first image as the second one. That's fine. But he went on to explain why the wire marked and crossed can be omitted. I did not quite get this part. This wire is connected at 3 junctions. Let us assume them P Q and R. He says that the current in top and bottom half of the junctions are equal. But who cares? If the current in junction P = Q = R, then there should be no current flow between P Q and R right? But how can we be sure about this?

I see A, B, ( what appears to be in lower case cursive) and R upper case printed. Where are P& Q at?

The reason why you can ignore any current flow through the lines that are crossed out is that by symmetry we know there is no current flow through that connection. See the attached sketch, which is for the connection into the central node. I've labeled each of the 6 leads to that central node A through F, and haven't bothered showing the resistors. From the symmetry of the circuit we know that whatever amount of current flows in from A must equal the amount that flows out through B, and whatever amount flows in through C must flow out through D, and whatever amount flows in from E also flows out through F. Therefore we can ignore the interconnections connections between A-B and C-D and E-F.

Attachment 48558

Oooooo! Got it.
And smoothy, I considered the junctions p, q, and r respectively. They are not in the picture. I considered them for the sake of discussion.
It is solved now. Thanks.