呼吸 is modified by 相手が打つときの, 「〝打つ〟の動作の〝う〟の頭をとらえて押し込む」というような, and 言葉にすると奇妙としか言いようのない
Correct. "There's certain 呼吸 that's like [ 相手が打つときの, 「〝打つ〟の動作の〝う〟の頭をとらえて押し込む」というような], which sounds peculiar when put into words.
〝打つ〟の動作の〝う〟の頭
This is where @naruto mentions the relationship to an existing question about "見鬼のけの字".
頭 here means the head, the front, the tip (as in the head of a nail). 「う」 is the first letter of 打{う}つ, and the phrase figuratively expresses the very first portion (頭) of the initial movement (う) of the strike (打つ の動作):
The head/tip/front of the (first letter) "s" of the motion of the strike
It seems that if you react to 〝打つ〟の動作の〝う〟, that's too late. You have to know the beginning of the beginning of the motion, probably even before the strike is started.
The rest: you have to recognise the "う"の頭. 押し込む here can probably be understood as "push forward" or "plunge". The actual action involved in 押し込む is not explicitly specified. It might mean the movement of the sword, your body or your spirit, but I am not sure. Probably some experience in Kendo, fencing or some other combat sport (which I don't have) might help feel the meaning.