Kelvin connection is useful (or a must have) in situations where you do precision measurements, and your measurement setup may influence the measurement results.
For example, you pass a current through a low-resistance device (a power MOSFET is a very good example, with 1-100 mOhm typical Rdson resistance), and you measure a voltage, to find the resistance R=V/I.
The voltage drop one the leads makes the voltage measurement on device impossible.
So, in Kelvin measurement setup, you use two pairs of probes - one pair is used to pass high (user-defined) current, and you do not care about voltage drop on these leads, and you use another pair or leads/probes (that you place close to the tips of the current forcing leads) to measure the voltage (the current through the sensing leads/probes should be low or zero).
Without a Kelvin setup, measuring resistance in the range below ~100 mOhm is very difficult (if not impossible).
When measuring such low resistance, be sure to eliminate the heating of the device.
Very often, pulsed, quick measurements are used for that.