Because in rooting media, the main objective is to induce the root which is being favoured by growth hormone and not the media salt concentration. Once the roots are induced, the half strength media is sufficient to sustain the required growth of the shoots.
Obviously, nutrient availability is less in half MS. Plant shoots will try to access more nutrients by inducing root formation. So half strength MS may serve as an additional induction of roots along with root hormones. Additionally, it may increase photosynthetic ability of shoots.
Additional to above answers, I would like to add that half strength MS media provide pre-acclimatization conditions for your obtained plantlets through adjusting the osmotic strength as well as adjusting the nutritional strength.
Because in rooting media, the main objective is to induce the root which is being favoured by growth hormone and not the media salt concentration. Once the roots are induced, the half strength media is sufficient to sustain the required growth of the shoots.
The mineral concentration of the culture medium affects rooting characteristics and for this reason some researchers have
proposed its reduction to half normal strength for rooting improvement since inorganic ions participate in processes regulating hormonal balances. The favourable effect of a diluted mineral solution on rooting can be explained by the reduction in nitrogen concentration.