The laws on "extremism" (first brought in in 2002, then amended in 2006 and 2007) are defined in the "The Federal Law on Combating Extremist Activity". However, it is up to the courts to interpret the law (except that swastikas are considered extremist without further interpretation). The law distinguishes between extremist activity (eg drawing a swastika on a wall) and encouraging extremist activity (publicly telling someone to draw a swastika) and punishes the latter considerably more harshly.
A translation of the full definition of extremism can be found in a briefing prepared for the EU by "SOVA Center for Information and Analysis" a Moscow-based Russian nonprofit organization that campaigns on racism and the misuse of extremism legislation, and which can be seen as having an "anti-Putin" stance.
The particular lines that affect Jehovah witnesses are
incitement to social, racial, ethnic or religious hatred;
propaganda of exclusiveness, superiority or inferiority of an individual based on his/her social, racial, ethnic, religious or linguistic identity, or his/her attitude to religion;
violation of rights, liberties and legitimate interests of an individual and citizen subject to his/her social, racial, ethnic, religious or linguistic identity or attitude to religion;
"Hatred" is also translated as "discord" in some sources. (I'm not a Russian speaker so can't comment on the nuance of the original.)
It is the role of the court to interpret these lines in the context of the activities of a particular group. The courts in Russia have on several occasions decided that the activities of the Jehovah Witnesses in Russia meet some or all of these criteria for being "extremist". The particular point that the court noted was that the evangelical activities of the church: going door to door and handing out evangelical literature was decided to "incite religious hatred".
The basic attitude taken by the court is that Jehovah Witnesses are a "cult" whose activities cause "serious public danger". And it is a reflection of this attitude that leads to the decision to liquidate the organisation as "extremist"
For the sake of clarity, I'm not arguing that the court was correct here or that the law is a good law. Nor am I arguing that JW are theologically "right", that would be off topic. I'm also aware that swastikas have a long noble history before the Nazis. I'm only attempting to answer the questions "What exactly are those laws, and how are they being applied"