Here "this tent" means the same thing as "the earthly dwelling that we live in." A groan is a sound that a person makes when they eagerly desire to have something that is good.
The words "our heavenly residence" means the same thing as "the place God will give us to live in." Paul speaks of the new body that believers receive after they die as if it were both a house to live in and a piece of clothing that a person can put on. (See: [[rc://en/ta/man/jit/figs-metaphor]])