보고 싶다/싶어 is literally 'I want to see.' Here, there is no object of 보다 but we can infer it from contexts.
If you were talking about a movie, it would be that movie. But if you show any Korean 보고 싶어 alone, they will understand it as '너를' 보고 싶다/싶어, I miss you. (I want to see 'you')
The difference between 싶다 and 싶어 was well explained by leelee88. But both are the informal(casual) expressions.