“You don’t have to take my word for it. Go see for yourself.” 「(私の言葉で)信じなくてもいいよ。自分で見に行きなさい。」

“I don’t have the letter anymore, so you’ll just have to take my word for it.” 「あの手紙はもうないから私の言葉を信じるしかない。」

Take their wordとは他に証拠がなくてもその言葉を信じるということです。しないなら、自分で確認してから信じます。

If you take someone’s word for something, you believe them without any other evidence that what they say is true. If you don’t take their word for it, you confirm by checking for yourself.