"Let's" is a contraction for "let us." It an invitation to someone to join you in doing something. "Let's go out to lunch" means "I don't want to eat by myself. I would like some company. Please join me. Let's go out and eat lunch together." "Let's hope" means "join me in hoping for something." It means the situation is uncertain, and you don't really know anything to do about it but hope, and you are inviting someone else to hope or wish along with you.

Hi. You can use this when you are hopeful or optimistic that something will happen, similarly to the use of "hopefully" or "God willing." You can put it in a sentence or use it as a response to someone else's statement. Let's hope the weather will be nice tomorrow for the wedding. I'm sure your big presentation for the boss will be great. (response) Let's hope!