It is used when someone looks like he's going to die / to cry / and so on.
- そうです = it seems that...
彼は泣きそうです。POLITE, CONCLUSIVE FORM
彼は泣きそうだ。PLAIN FORM
He is about to cry. He's almost crying. He's at the point of crying.
彼は死にそうです。POLITE, CONCLUSIVE FORM
彼は死にそうだ 。PLAIN FORM
He is about to die. He's almost dying. He's at the point of dying.