"Please forgive what may be a stupid question, but why is it necessary to create temperatures hotter than the Sun's core?"
The rate of energy production at the Sun's core is quite astonishingly low. Wikipedia notes "At the center of the Sun, fusion power is estimated by models to be about 276.5 watts/m3.". That's about a tenth of a kettle from a volume that would hardly fit on your kitchen worktop. (The same article, https://en.wikipedia.org/wiki/Solar_core, notes that it then takes about 170,000 years for the energy to escape, but that's not relevant to the current discussion -- merely a fun factoid.)
I imagine that most physics graduates would guess a much larger figure, so your question is not stupid at all.
As for the answer, I would imagine that by raising the temperature you raise the reaction rate exponentially. (That's how these thermodynamic scenarios usually play out.) Therefore, to achieve your (correct) objective of a self-sustaining rate of energy production at the lowest temperature, you need to be a little hotter.
The plasma, by the way, forms at a few *thousand* degrees because plasma is just matter so hot that the electrons fall off.
Biting the hand that feeds IT
