Example, assume you have space ship traveling at 0.5c relative to an observer.

The spaceship drops a buoy and then accelerates away from it at 0.5c in the same direction as the relative motion it already has with respect to the observer.

Using non-relativistic velocity addition, you would expect the ship to now have a relative speed of 0.5c+0.5c= 1c with respect to the observer.

Instead, he ends up with a relative speed of

\frac{0.5c+0.5c}{1+\frac{0.5c(0.5c)}{c^2}}= 0.8c

In other words, even though the buoy still moving at 0.5c relative to the observer, and the ship moving at 0.5c relative to the buoy, the ship is only moving at 0.8c relative to the observer.

The ship can try it again; dropping a new buoy and accelerating away at 0.5c. After which the ship will be moving at 0.5c relative to the second buoy, at 0.8c relative to the first buoy, and

\frac{0.5c+0.8c}{1+\frac{0.5c(0.8c)}{c^2}}= 0.92857c