The web-of-trust argument is good but raises the question of why he said:
I taught him! I thought I knew him!
You have to remember that Snape is primarily —like Slughorn— a potions expert. That suggests that he did potions to the highest level at Hogwarts, under Slughorn's tutoring.
Snape was also a Slytherin, the house Slughorn was the head of. The two would have had lots of contact at Hogwarts. It's possible that Slughorn considered him a protégé... In a way more intimate than he did with his other "collected students".
You also have to factor in the timings of things. To the right sort of person, being a Deatheater seemed like a positive thing. The Blacks' early allegiance to Voldemort's cause. It's possible that Slughorn shared the creed that magic is might.
It wasn't until later that the killings were overt and the Ministry started prosecutions. And it was at that point Snape was turned and publicly trusted by Dumbledore.