It arises as the variation of the Einstein-Hilbert action
S = ∫ (1/2κ) R √|g| d⁴x
with respect to the inverse metric
δS = ∫ (1/2κ) G_μν δg^μν √|g| d⁴x
As to what other geometric meaning the tensor has, apart from its application here and to the Einstein field equations: both it and its generalizations that include the cosmological coefficient, are used to define Einstein manifolds. These are manifolds in which the Ricci tensor has a fixed ratio to the metric tensor.
A couple notes on the coupling coefficient κ are in order here, to correct a widespread error that I see being repeated in some of the replies here.
Up to powers of light speed, c, the 8π factor in the expression κ = 8πG is specific only to 4 dimensions and is related to the 4π that appears in the formula for the surface area of a unit 2-sphere and the 4π/3 for the volume of the solid 2-sphere. For n > 3 dimensions it generalizes to (n-1)(n-2)V/(n-3), where V = √(πⁿ⁻¹)/((n-1)/2)! is the volume for the solid (n-2)-sphere, where the convention (-1/2)! = √π is used.
Second, the powers of c come directly out of the action formula by dimensional analysis. Using M, L and T respectively to denote dimensions of mass, length and time, the dimensions are given, for n-dimensional space-times, by
[S] = ML²/T = [h]
[R] = 1/L²,
[√|g| dⁿx] = Lⁿ,
[G'] = Lⁿ⁻¹/(MT²) = the n-dimensional version of Newton's constant G,
[c] = L/T,
where h is Planck's constant. Therefore,
[hκ] = [∫ R √|g| dⁿx] = Lⁿ⁻² = [hG'/c³]
the dimensions of the Planck n-2 area for n-dimensional space-times ... or (for n = 4) the Planck area. Therefore,
[κ] = [G'/c³], not [G'/c⁴]
and the correct expression for κ for dimensions n > 3 is
κ = (n-1)(n-2)/(n-3) √(πⁿ⁻¹)/((n-1)/2)! G'/c³
which for the case n = 4 (and G' = G) reduces to
κ = 8nG/c³.
The literature is interspersed with powers 2, 3 and 4 (e.g. Einstein used 2 in The Meaning of Relativity), the expression κ = 8nG/c⁴ comes by way of a faulty dimensional analysis which fails to properly distinguish tensors from tensor densities (e.g. the mass density ρ and pressure p are components of the stress tensor density, not the stress tensor) and/or to properly account for the dimensions of the stress tensor itself.
This is discussed in further detail in a recent thread under sci.physics.research - the standard forum for physics-related questions.
The issue of the numerical factor is also discussed here by Mansouri et al.: https://arxiv.org/abs/gr-qc/9609061