Noun and の adjective
みぎきき
わたしの息子は右利きではありません。
My son is not right-handed.
ぼくは左利きなのに、お母さんが間ちがえて右利きのグローブを買っちゃったんだ。
Although I am left-handed, my mom bought a right-handed baseball glove by mistake.
彼女のこと、すごく好きだったんだけど、右利きだって分かってから冷めちゃったんだよね。どうしてかは分からないけど。
I really liked her but I was turned off when I found out that she was right handed. I don’t know why.
The right side of your body is the side you profit most from. So, you use it a lot more. That's because you're right handed.
Both kanji use the kun'yomi vocab readings. So, think to the words 右 and 利く to learn how to read this one. It probably reads like this because it's two separate words come together (right + handed) rather than one word.