I found this sentence:
I got the meaning of the first part, but I don't get the exact meaning of 気になりだした.
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I think it comes from 気になる which means to worry. In Japanese it is common to combine two verbs to form a new one. 着る(to wear) + 替える(to switch/replace) = 着替える(to change clothes). The preceding verb must be changed to the verb stem before being attached to the other verb. 気になる = 気になり. Then the verb だす is attached. If it's correct then it should mean... To start worrying. I'm still learning so I cannot guarantee my validity.