Aqueous solvated hydrogen ions (acid) and hydroxide ions (base) both have especially high conductance in water. So when you have the neutral solution with not much of either, conductivity drops.
Suppose you are on one side of a crowd and you have a baseball that you want to deliver to the other side. You can't throw the hard ball because that is too dangerous, so you have two choices. You can muscle through the crowd, carrying the ball along, or you can pass the ball to the next person, then the next, etc while you stay put. Most likely the second approach is faster.
Now suppose you have a hydrochloric acid solution. The chloride ions can move only slowly by muscling through the crowd. But the hydrogen ions don't have to do that. The water molecules have multiple dissociable protons, so instead of moving a whole solvated proton structure through the mass of liquid it's possible to just pass an individual proton from one molecule to the next. The bulk of the water structure stays in place like you do when you pass the baseball. The hydrogen ions in the acid pass the "baseball" of positive charge instead of having to muscle through the liquid crowd.
The same thing happens with an alkali like sodium hydroxide. While the sodium ions are fighting their way through the liquid, the hydroxide ions efficiently relay their charge from one molecule to another, by taking protons from neighboring water molecules.
So solvated hydrogen ions in aqueous acid and hydroxide ions in bases can conduct their charge very efficiently by exploiting their proton-transfer acid-base reactivity with water. The ions that remain when the acid and base are neutralized to form a stable salt can't do that and thus conduct electricity only slowly.
For an animation of what happens in acid, see https://en.wikipedia.org/wiki/Grotthuss_mechanism.