Initially, the Tanners weren't very happy to house a short smartass who wanted to eat their cat, but they soon realized the gravity of the situation after a man from the Alien Task Force showed up at their door. Curious, Kate Tanner asked the man just what would happen to ALF if they did catch him.
Actual quote from the first episode:
"We'll see how it responds to intense heat, freezing cold, high voltage, toxic substances, pain, sleep deprivation, inoculation [that's needles], and, of course, dissection."
"Why don't you just pull its toenails out?"
"You didn't let me finish."
Forget trying to learn about advanced science from this obviously technically superior species, our government would rather see what he looks like on fire.
Naturally, the Tanners decide that annoying or not, they absolutely have to keep ALF safe from the Alien Task Force so he won't be tortured to death. Over the next four years, the Tanners learned to love ALF and accept him as part of the family, until the Alien Task Force finally caught him!
The final shot of the entire series.