The idea of extra-dimensions is not new ! It first popped out in the context of general relativity in 1921, when Kaluza proposed to extend the Einstein equations to 5 dimensions, with a compact extra-dimension (Kaluza-Klein model). The resulting 4-dimensional model contains GR and electromagnetism (Kaluza-Klein reduction). While this looked like THE unification of GR and electromagnetism, the proton was not stable in this model. This procedure was forgotten until string theory was formulated. For consistency, string theory requires 11 dimensions (depending on the model, but always more than four). Our usual 4 dimensions are recovered as a low energy limit of string theory, using the principle of the Kaluza-Klein reduction (in a more sophisticated way). Therefore, a collection of fields appears in the 4 dimensional effective theory.
A phenomenological way to approach the higher dimensional models is to truncate the reduced model (not always consistently… it is a phenomenological approach, so it is inspired from basic principles) and to look at gravity alone in more than 4 dimensions. The model can then be refined with additional matter fields, describing our usual matter and interaction particles. Note that all interactions except gravity (because it is the spacetime geometry and now the spacetime is higher dimensional) should propagate in our 4 dimensions only. This leads to the concept of braneworlds.
In the braneworld scenario, usual matter fields (fermions and gauge bosons) are confined in a 4 dimensional submanifold of a higher dimensional manifold. Again, gravity affects the whole manifold since it deals with its geometry, while the three other interactions are confined on the brane by some mechanism to be understood.