One of the reasons inversions are useful is that they enable you to find chord shapes which are close together on the fretboard, as demonstrated in the following example which uses various inversions of the chords D, G and A on the 2nd, 3rd and 4th strings.
By using inversions which are close together on the fretboard you can create a smoother transition between chords. The way the individual notes connect in chord changes is called voice leading.