Felix,
here is a version that works - DiscretizeRegion[]
often saves the day!
f = {Indexed[#1, {1}]/(
Indexed[#1, {1}]^2 + Indexed[#1, {2}]^2 + Indexed[#1, {3}]^2),
Indexed[#1, {2}]/(
Indexed[#1, {1}]^2 + Indexed[#1, {2}]^2 + Indexed[#1, {3}]^2),
Indexed[#1, {3}]/(
Indexed[#1, {1}]^2 + Indexed[#1, {2}]^2 + Indexed[#1, {3}]^2)} &;
planeRegion = DiscretizeRegion@ImplicitRegion[z == 5, {x, y, z}];
tr = TransformedRegion[planeRegion, f];
Region[tr]