Step 1: state the law in plain words.
Kohlrausch's law of independent migration of ions says that when an electrolyte is spread out to infinite dilution, the cation and the anion each carry current on their own without disturbing one another. So the limiting molar conductivity of the whole electrolyte is just the cation's share plus the anion's share:
\[ \Lambda^\circ_m = \nu_+ \lambda^\circ_+ + \nu_- \lambda^\circ_- \]
where each ion brings its own fixed contribution.
Step 2: the easy case of a strong electrolyte.
Picture a graph of molar conductivity against the square root of concentration. For a strong electrolyte such as KCl, the ions are already fully free, so as we dilute it the points fall along a gentle, almost straight line. We simply extend that straight line back to zero concentration and read off the value where it meets the axis. That value is $\Lambda^\circ_m$.
Step 3: why a weak electrolyte refuses this trick.
A weak electrolyte like acetic acid is hardly split into ions at ordinary concentrations, and it only ionises fully when it is enormously diluted. Because of this, its curve does not run straight; near very low concentration it suddenly shoots steeply upward. There is no neat straight portion to extend, so trying to extrapolate gives a useless guess. Instead we find its $\Lambda^\circ_m$ indirectly, by adding up the known ionic conductivities using Kohlrausch's law.
The weak-electrolyte curve bends sharply, so we use ion values rather than extrapolation.