믿다 - to believe, to trust (in something, someone)
믿을 수 없어요 = "I can't believe it"
보다 - [in this context] to try. --> i find sometimes 보다 doesn't always translate into English as a strong meaning of 'try'. For instance, to try something is a stronger indication of an attempt than 'give it a shot' 'we will see', which is softer.
믿어 보자 =
1. Let's try to believe (in something, someone)
2. Let's believe (in something, someone)
I'm not a native speaker, this is what i think.
Hope it helps.