Why does the conductivity of an electrolyte change when we dilute it?

The two quantities measure different things, so they move in opposite directions. Specific conductivity (conductivity per unit volume) decreases on dilution because as you add water, the number of current-carrying ions in each unit volume falls, so a given cubic centimetre conducts less. Molar conductivity, however, is the conductivity of all the ions coming from one mole of electrolyte, no matter the volume. On dilution, a weak electrolyte ionises more and the ions move more freely with less interference between them, so the total contribution per mole increases. So both observations are correct: per volume, conductivity drops because ions are spread thinner; per mole, it rises because more ions are free and active. Keep clear which basis, volume or mole, each term uses and the contradiction disappears.