Memory Alpha doesn't explain this at all - just acknowledges that the Voth did indeed originate from Earth. I would assume a spacecraft of some sort was essential, but this leaves open multiple options. Eg, does that mean the Voth actually were the first earth species to invent the warp-drive if they used a warp-capable ship, or did they travel via an asteroid or something like that? My question is how did the Voth get into the Delta quadrant in the first place - did they use a warp-capable spaceship, or a spaceship at all? (I don't know if there's some reference in a novel or something)
The Voth are capable of high-warp capabilities above the standard warp factor speed (1 - 9.99), technically they did not use the warp drive to propell them across the galaxy as anything above Warp 10 is defined as anywhere in the universe.
It can be concluded that they may have come across a wormhole however that would be unlikely as they would need to move a whole species within a certain timeframe before the wormhole is deemed unstable and collapses.
This is pure speculation as I have failed to find any sources containing this data, they created a propulsion similar to the slipstream drive or the transwarp drive.
Source 1 - Maximum warp velocity across the series
Source 2 - Quantum slipstream and the similarities of the concept relative to the Borg transwarp drive.
Source 3 - Wormhole the Ferengi used to travel between Alpha Quadrant and the Delta Quadrant.
Source 4 - Voth's past, evolving technologically faster and becoming superior but leaving to avoid destruction.