roygilbo AppVerifier's mission is to check whether an APK has a valid signature. It knows the correct signer for some apps, but not all (in which case it is necessary to provide it key information so it can run the check). It is useful for detecting fake versions of APKs.
AppVerifier absolutely cannot prove that a given APK won't do something bad. Proving that programs don't do bad things is very difficult, and there are some bad things that, mathematically speaking, it is not possible to prove that a program won't do (see Halting Problem).