I can understand your confusion. Here's what I know: This is a typical American English construction. In American English, it is common to hear people combining the verbs 'come' and 'go' with the verb indicating the reason why the person comes or goes to this place e..g. 'Come eat' or 'go see'. In British English, we would say, for example, either 'Come and eat your breakfast' (to show sequence) or 'Go to see your doctor' (to show purpose). We wouldn't normally put the two verbs together. Here's what I don't know: I'm not sure whether speakers of AmE would regard 'come eat' or 'go see' as a grammatically correct construction or as an informal colloquialism. Perhaps someone from the US or Canada could tell us.

You can say 'I want to go to see the film' or you can say 'I want to go and see the film'. 'I want to go see something' would be used more in spoken language rather than in written language and is more informal.