At this question, abstract interpretation has a nice in-depth look. However, I'd like someone to clearly and very precisely state how abstract interpretation takes the result of Rice's Theorem over the Turing-complete programming languages/Turing machines, and uses abstract interpretation to get similar results for decidability for non-trivial program properties; or, rather, how does abstract interpretation get around Rice's Theorem/the Halting Problem?
To be as clear and concise as possible, I'd try saying this
Abstract interpretation may only over-approximate properties of programs: the most precise abstract value of a program $P$ may be $a$, but any algorithm for computing an abstract value will in general compute some value $a'\geq a$ for $P$ (possibly $\top$ in the worst case).
This is logically redundant with the answers already given by Cody and Mikolas, but here's an intuition that might help. The property that any abstract interpretation proves about a program is guaranteed to be correct, but sometimes this is because the answer it brings back is "I don't know". That answer may be useless, but it isn't wrong. Basically, if you want a decidable version of a decision problem, you have to add this third option, so that the possible results are not "yes"/"no", but "yes"/"no"/"I give up".