The phrase implies the person's presence is intrusive or uncomfortable for others, especially a couple. When I realized they wanted privacy, I felt like I was playing gooseberry.
Pro Tip 2/3
Romantic Context
Often used when a couple wants to be alone, and a third person is present, making the situation awkward. Hanging out with them on Valentine's Day made me the gooseberry.
Pro Tip 3/3
Idiomatic Usage
Understand it as an idiom, not to be taken literally. 'Gooseberry' doesn't refer to the fruit in this context. I declined the invite, not wishing to play gooseberry at their anniversary dinner.