Context: Vegeta sent his underling to fight Kakarot. His underling lost. Vegeta is perplexed.
nasakenai yatsu da なさけないやつだ･･････
[What a] pathetic guy.
Referring to a guy who just lost to Kakarot.
masaka {ano Kakarotto aite ni kono ore ga wazawaza ugoku} koto ni naru towa na...... まさかあのカカロットあいてにこのオレがわざわざ動くことになるとはな･･････
[It's unbelievable that] it ended up in such way that {against that Kakarot, I have to [fight]}.
In other words: Vegeta didn't expect he'd have "to move," ugoku 動く. He expected his underling to handle it. So he wouldn't have to act, he wouldn't have to fight Kakarot.
However, since his underling lost, the "situation became in such way that" it's necessary for him to make a move.
That's what koto ni naru is expressing: the situation ended up that way.
