How is 我简直不敢相信！ translated as "I can't believe it!"
我簡直不敢相信 translates literally to I simply don’t dare to believe [it]. You can translate this to English using I can’t believe it!, with an exclamation mark at the end to capture the mood. You say this when something outrageous or ridiculous has happened.

我不能相信 sounds like I’m unable to believe [it], and doesn’t really capture the same mood. You may say this if you’re refusing to believe something, maybe because of lack of evidence.