The meaning is similar to "[engineers and personnel] who work for ...". Note that the participle working refers not to personnel only, but to engineers and personnel. Let's rephrase:
There's been a "renaissance" of engineers and personnel who work for Russia's defense industry, he says, an elite class in the Soviet Union that fell apart in the chaos of the early post-Soviet period. Now they're back, and the Kremlin is paying attention. The group has almost nothing to do with the market economy, Lukyanov says, and they support Putin.
Engineers and personnel of what kind? - Of the kind that works for the defense industry.
Following the dissolution of the USSR, there was a lack of funding, and these engineers and personnel suffered from low salaries and the necessity to look for other jobs. Now Putin's government is focused on restoring this sphere: manufacture of armaments, military vehicles, etc. Thus, these engineers and personnel who work for the defense industry undergo 'a renaissance'.
"Working for Russia's defense industry" is a present-participle phrase connected to "engineers and personnel". It is also called 'participial phrase'.