It is a phrase that means “to swallow“.
It is used to describe situations in which you accept something without question. It is usually used in a negative way. “鵜呑みにしない“ is antonym for “鵜呑みにする“ and it is similar to “話半分に聞く(*1)“ .
Ken and Leo are talking about how long it will take for Tom to finish his report.
- Ken: I’m waiting for Tom to finish his report. Do you know where he is now?
トムがレポートを終えるのを待ってるんだ。どこにいるか、知っている？ - Leo: Yes, he left about 30 minutes ago. He told me that there were 10 slides of his report left to write and he would be done with them until tomorrow afternoon.
あぁ、30分前に帰ったよ。10ページ残っているけど、明日の午後までに終わるってさ。 - Ken: Hmm, I don’t think he will be able to finish on the time he planned. He always overestimates his ability to do his tasks.
ふんー、アイツが言った通り終わるとは思えないんだけど。いつも見積もりが甘い(*2)んだよ。 - Leo: It was so stupid of me to believe what he said!
アイツの言った事を鵜呑みにした俺がバカだった！
Kumi invited Meg and Amy to the party.
- Meg: I’m so excited for the party tomorrow. We could make friends, right?
明日のパーティーが楽しみだな。友達作れそうだね？ - Amy: No, and I’m not going to that party. Last time, I was so excited to talk to nice guys at the event she set, but I was there only to serve drinks to the guests.
無理だと思うし、私は行かないよ。前回、彼女が招待してくれたイベントで、素敵な人に会えると思って楽しみにしていたのに、ずっと飲み物を配らせられたもん。 - Meg: Really? This time, she told me that she would be glad if I handed out drinks to the participants at the beginning of the party. OMG! If I bought into what she said, I would work the whole night!
本当？今回、パーティーの初めにドリンクを配ってほしいって言っていた。(ヤバ！)彼女の話を鵜呑みにしたら、一晩中働かなきゃいけないね！ - Amy: You might want to turn down the party.
(パーティーを)断った方がいいかもね。
*1 話半分に聞く: It means to take something with a grain of salt. If you’re interested in more detail, please click HERE.
*2 甘い: It means to be over-optimistic. If you’re interested in more detail, please click HERE.