6 ideas
A gene's phenotypic effects are not confined to the body it sits in but include all consequences it has on the world, such as a beaver's dam or a caddis fly's protective case. Selection acts on these external structures exactly as it acts on bodily traits, because they too are products of genes competing for representation in future generations.
Organisms and bodies are temporary vehicles, but the replicating gene is what persists across generations and therefore what natural selection ultimately optimizes. Shifting the unit of analysis from organism to gene resolves puzzles where a trait benefits the gene at the expense of the body carrying it.
A parasite's genes can shape the behavior and body of its host to serve the parasite's reproduction, meaning the host's altered phenotype is partly an expression of foreign genes. This proves phenotypic effects can cross the boundary of the individual organism entirely.
An animal can be made to act against its own interest when another organism's genes hijack its nervous system or behavior, as when a cuckoo chick's gape compels a host bird to feed it. The exploited behavior is a phenotypic effect of the manipulator's genes operating in another creature's body.
Replicators are entities of which copies are made (genes), while vehicles are the survival machines (organisms) that carry replicators and interact with the environment. Clarifying which role an entity plays prevents the confusion of treating the body as the thing being preserved when it is only the gene's temporary instrument.
There is no principled biological reason to draw the boundary of selection at the skin of an individual; the choice is a convention, not a fact about how genes propagate. Once you stop privileging the organism, behaviors and structures that span multiple bodies become legitimate targets of evolutionary explanation.