While a lot of it will be eventually achieved due to orbital mechanics, most of the push was given to it by the Delta IV Heavy. For those who may not be familiar, the launches are always spectacular due to the pre-flushing of Hydrogen through the system. NASA has an HD video that you can download, https://www.youtube.com/watch?v=AlyuSwRSVHU
The PSP's speed is impressive without context, but it's even more impressive in context. Here are phenomena in the same range as the PSP.
The Earth rotates around the sun at a speed of,
29,800 m/s or 0.01% the speed of light
Our galaxy approaches the Messier 98 https://en.wikipedia.org/wiki/Messier_98 galaxy at a speed of, 140,000 m/s or 0.04% the speed of light.
PSP is faster than both of these at, 163,000 m/s or 0.05% the speed of light.
Due to the quirks of orbital mechanics, by the end of its life, the probe will be even faster and it will reach, 192,000 m/s or 0.06% the speed of light
It's 8,000 m/s shy of our solar system's orbital velocity around the galactic center.The design is a stunning accomplishment of engineering and daring. If you would like a detailed overview of how the mission trajectory was designed, the navigation team has written an excellent explainer that you can read here, https://trs.jpl.nasa.gov/handle/2014/47575