What is the nature of reaper destroyer weapons?
This actually depends on what kind of Reaper you're talking about, as they're classified into four known types. I'm assuming you're actually asking about Sovereign class dreadnought ships, also known as Reaper Capital Ships, rather than Destroyers, as those are a smaller subtype. Sovereign's primary weapon in particular is described here on wiki page:
Sovereign’s destructive power was unrivaled in the known galaxy. Each of the "tentacles" extending from its bow was equipped with a powerful "magnetohydrodynamic" weapon which ejects a stream of molten metal at a fraction of the speed of light which could tear through a cruiser in a single sustained burst. Its gigantic spinal-mounted gun was able to rip through the hulls of even the largest of dreadnought-class ships with ease, effortlessly penetrating their kinetic shields.
If you want official citations from the game itself, this information is detailed more in the ingame Codex entry from Mass Effect 2 in the Thanix upgrade to the Normandy SR-2, a section of which I have quoted here:
Contrary to popular belief, Sovereign's main gun was not a directed energy weapon. Rather, its massive element zero core powered an electromagnetic field suspending a liquid iron-uranium-tungsten alloy that shaped into armor-piercing projectiles when fired. The jet of molten metal, accelerated to a fraction of the speed of light, destroys targets by impact force and irresistible heat.
More information can be found under the Reaper Capabilities Codex entry:
The main gun on a Reaper capital ship dwarfs that of the Alliance's Everest-class dreadnoughts. No dreadnought has yet survived a direct hit from the weapon. Estimates put its destructive power anywhere from 132 to 454 kilotons of TNT. Even if the target is hardened, as in the case of a surface-based missile silo, the gun can instead bury the target beneath molten metal. Precise targeting computers and correctors also give the Reaper weapons a longer effective range than organics' dreadnoughts or cruisers.
In regards to the other subtypes, Destroyers are smaller than the Capital Ships and possess less firepower than Capital Ships, but some of them possess a Hades cannon, which is described as a "massive directed energy cannon" used as an anti-aircraft weapon. These were seen in ME3 during, mostly during Earth's occupation.
It's a high energy particle weapon. Given that the Thanix Cannon is reverse engineered from Sovereign's guns, and the destroyers would probably be using the same tech. It would work as such:
The Thanix's core is a liquid alloy of iron, uranium, and tungsten suspended in an electromagnetic field powered by element zero. The molten metal, accelerated to a significant fraction of the speed of light, solidifies into a projectile as it is fired, hitting targets with enough force to pierce any known shield or armor. The gun can fire reliably every five seconds. The weapon's relatively small size allows it to be mounted on most fighters or frigates, including the Normandy SR-2, and gives them firepower rivaling cruisers.
From the wiki