N3
のは~の方だ
のは~のほうだ
It is...that..., It is...who..., ...is the one that..., ...is the one who...
言い出したのは俺の方だから、俺がなんとかしておくよ。
I am the one that said it, so I will do something about it.
Explanation
「~のは~の方だ」 compares two sides and stresses 'actually it's this side'. It is often used to turn things around and take the blame or credit yourself: 「謝らなければならないのは、私の方です」 (I'm the one who should apologize), 「迷惑をかけたのは僕の方だよ」 (I'm the one who caused trouble). 「のは」 brings up the topic, and the focus goes on 「~の方だ」.
Mixed up with
When the other person apologizes or thanks you, this pattern says 'no, no, it's me who should...'. It sounds humble and considerate.
「~のは私だ」 alone just flatly points out 'it's me'. Adding 「の方」 gives the sense of weighing the two sides and leaving the other person at ease.