In layman's terms why is the time component of the stress-energy tensor associated with energy density? Why is the time component of the stress energy tensor $T^{00}$ associated as energy density?
 A: Consider a stationary box of volume $V$ filled with total electric charge $Q$.  The charge density of the box is $\rho_0 = \frac{Q}{V}$, and the box is stationary so the current density is zero.  If you boost to a frame moving with relative velocity $-\mathbf v$, then the volume of the box will be contracted, and so the charge density will increase; the new charge density will be $\rho = \gamma \rho_0$, where $\gamma \equiv (1-v^2/c^2)^{-1/2}$.  At the same time, the observed current density would become $\rho \mathbf v = \gamma \rho_0 \mathbf v$.
The charge density and current density transform like the time and space components of a 4-vector, so we introduce
$$\mathbf J = (c\rho,\rho \mathbf v)$$

Now consider a stationary box of volume $V$ filled with mass $m$.  The energy density of the box is $u_0=\frac{mc^2}{V}$, and the box is stationary, so the momentum density of the box is zero. If you boost to a frame moving with relative velocity $-\mathbf v$, then the volume of the box will be contracted, and the energy of the box will increase, so the energy density will increase twice; the new energy density will be
$$u = \frac{\gamma mc^2}{V/\gamma}=\gamma^2 u_0$$
and the new momentum density will be $\frac{1}{c^2}\gamma^2 u_0 \mathbf v$.
Energy density and momentum density (unlike charge density and current density) do not transform like the components of a 4-vector.  They do, however, transform as though we defined a (2,0)-tensor with components
$$T^{\mu \nu} = \pmatrix{u_0 & 0 & 0 & 0\\0&0&0&0\\0&0&0&0\\0&0&0&0}$$
and performed a Lorentz transformation on it.  Let $\mathbf v = v\hat x$, so
$$\Lambda^\mu_{\ \ \nu} = \pmatrix{\gamma & \beta\gamma & 0 & 0 \\ \beta\gamma&\gamma&0&0\\0&0&1&0\\0&0&0&1}$$
We then have
$$\Lambda^\mu_{\ \ \rho}\Lambda^\nu_{\ \ \sigma}T^{\rho\sigma}=\Lambda^\mu_{\ \ 0}\Lambda^\nu_{\ \ 0}u_0 = \pmatrix{\gamma^2 u_0& \frac{1}{c}\gamma^2 u_0v&0&0\\ \frac{1}{c}\gamma^2u_0 v& \frac{1}{c^2}\gamma^2 u_0v^2& 0 & 0\\0&0&0&0\\0&0&0&0}=T'^{\mu \nu}$$
$T^{00}$ is identified with energy density, while $T^{0i}=T^{i0}$ are identified with momentum density (times $c$).  The physical interpretation of the $T^{ij}$ term is not so obvious; its interpretation is borne out when one performs the same analysis on a relativistic fluid, in which case the $T^{ij}$ terms are identified with the fluid's stress tensor.


What why is energy density in the time component of the tensor? Or to put it another way what does energy density have to do with time?

Energy has a very deep connection with time.  In this case, it is the $00$-component of the stress-energy tensor ultimately  because it is the $0$-component of the 4-momentum.
The deeper explanation for this is that energy is the conserved Noether charge associated with time-translation symmetry.  If a theory with Lagrangian $L$ is invariant under spacetime translations $x^\mu \rightarrow x^\mu + dx^\mu$, then a quantity $p^\mu$ (the 4-momentum) is conserved; $x^0$ is time and $p^0$ is energy.  The continued association between energy and time extends to the stress-energy tensor.
