I would think that everything slows down, maybe not so much in the vacuum of space, but why would these particles continue accelerating?
Well, there is no such thing as a perfect vacuum, first of all. All of the space surrounding the Sun is filled with an ionized gas -- called a plasma -- that flows out from the sun at supersonic speeds. This supersonic wind is called the solar wind.
Sometimes the sun emits large blobs of plasma called coronal mass ejection (CME). If these propagate much faster than the solar wind, they can generate a shock wave on their leading edge. These are not hydrodynamic shocks, but magnetized shocks. Magnetized shocks can accelerate/energize charged particles to extremely high energies (e.g., see discussion and references at https://physics.stackexchange.com/a/618127/59023).
So as to why the particles get accelerated, it's due to a CME shock running into the termination shock of the heliosphere. You can think of this like two walls moving closer together and between them is a ping pong ball bouncing off of them. If the walls move together quickly, the ping pong ball can gain a lot of momentum/energy.
He says similar processes have been observed within the borders of our Solar System where solar wind is most powerful.
This is just PR-related phrasing that is not conducive to actually conveying relevant information. The solar wind is more dense closer to the sun but by the orbit of Mercury, the solar wind speed is relatively constant all the way out past Pluto.