The principle of operation of Geiger meter (have a look here) is based on ionization events. The electrons in Geiger–Müller tube need to get enough energy to become capable of creating electron-ion pairs. The creation of those pairs in significant amounts causes a current pulse to flow from the anode to the cathode in G-M tube. The current is proportional to what you read on Geiger meter.
Now the question becomes, where do electrons get their energy from to start the ionization process. In nuclear reactions, gamma rays are emitted which are highly energetic, so the electrons in G-M tube become energized and you get a spike in meter's reading.
Plasma balls don't emit gamma rays or even X rays (have a look here). However, they emit some other forms of radiation which is still capable of energizing electrons to create electron-ion pairs. You can test that yourself at home by making an experiment like the one in this question.
So briefly, you get this spike in reading for the same reason a fluorescent light bulb lights when it is put close to the plasma globe.
The question whether how strong those radiations are, how close you need to be to be able to detect them, and whether it is dangerous or not, all those questions are addressed in this paper. As dmckee said, there is no harm for a human to be that close to it, but there is strong enough radiation coming out of plasma globe to be detectable by some means. Examples of those means are, Geiger meter, fluorescent light bulbs, and Multimeters as shown here
Hopefully, that was helpful.