She hesitantly gave her opinion.
To act in a reserved manner.
内気 means 'shy' as a personality trait, while 遠慮がち emphasizes holding back out of consideration for others.
遠慮がち describes someone who is reserved and hesitant out of consideration for others, the opposite of the boldness implied by 無遠慮.
Standard kanji spelling.
Alternate kanji form using 勝ち; less common.
Compound of 遠慮 (restraint, diffidence) + がち (tendency to).