Definition 1. A spacetime is said to be spatially homogeneous if there is a one-parameter family of spacelike hypersurfaces $\Sigma_t$ foliating the spacetime such that for each $t$ and for any points $p,q\in\Sigma_t$ there is an isometry of the spacetime metric $g$ which takes $p$ to $q$.
Definition 2. A spacetime is said to be isotropic if at each point there is a congruence of timelike curves, with tangents denoted $u$, satisfying: Given any point p and two unit spacelike vectors in $T_pM$, there is an isometry of $g$ which leaves $p$ and $u$ fixed but rotates one of these spacelike vectors into the other.
Restrict $g$ to a Riemannian metric $h$ on $\Sigma_t$. The geometry of each "leaf" of the foliation must inherit homogeneity and isotropy.
Let ${}^{(3)}\operatorname{Riem}$ be the Riemann tensor on $\Sigma_t$, $R_\Sigma$ be the scalar curvature and $T$ be the tensor field $$T(X,Y)Z=6\left[h(Z,Y)X-h(Z,X)Y\right]$$
for vector fields $X,Y,Z$.
Theorem. Homogeneity and isotropy of $\Sigma_t$ $\Leftrightarrow$ ${}^{(3)}\operatorname{Riem}=R_\Sigma T$, $R_\Sigma=\text{const.}$
Proof. Construct the Riemann tensor of $\Sigma_t$ using $h$. One may view this as an endomorphism $L$ of the space of $2$-forms $W$. By the symmetry properties of the Riemann tensor, $L$ is symmetric, and by a theorem in linear algebra, $W$ has an orthonormal basis of eigenvectors of $L$. If the eigenvalues were distinct, one could pick out a preferred $2$-form on $\Sigma_t$. Using the Hodge star on $\Sigma_t$, one could then construct a preferred vector. Since this would violate isotropy, the eigenvalues must be equal. We call this value $K$:
$$L=K\operatorname{id}_W$$
In other words,
$${}^{(3)}R_{ab}{}^{cd}=K\delta^c{}_{[a}\delta^d{}_{b]}$$
where ${}^{(3)}R_{ab}{}^{cd}$ are the components of ${}^{(3)}\operatorname{Riem}$. Contracting everything appropriately gives
$$R_\Sigma=3K$$
Homogeneity automatically fixes $K$ to be a constant. $\quad\Box$
This proof very closely follows the one given in Wald, R. M. 1984, General Relativity (Chicago University Press).
Actually there are some equations and results that make the Weyl tensor more intuitively clear, and that also makes the intuitive description that it denotes the deformation of a sphere, whereas the trace of the Riemann tensor is more closely related to its volume growth.
That deformation of the shapes, is due to the tidal forces, and more technically to the shear in the spacetime. The Weyl tensor describes it.
First is the fact that it is also called the conformal tensor, or the conformal Weyl tensor. That is, for two metrics that are conformally related, i.e., $g' = fg$, with $f$ a conformal transformation function of the spacetime coordinates, then, $C' = C$, with $g$ the metric and $C$ the Weyl tensor.
That means that a conformal (or simplistically) 'volume' change leaves the Weyl tensor invariant. The transformations or variations which affect the Weyl tensor are more the deformations than the 'volume' changes. This is explained a little better in Wikipedia at https://en.m.wikipedia.org/wiki/Weyl_tensor.
But one can do better, and understand what it denotes in spacetime.
In essence, the Weyl tensor describes the shear of null geodesics.
Thus, how geodesic balls deform. From a nice description in
https://www.physicsforums.com/threads/geometrical-meaning-of-weyl-tensor.708383/
and with mostly a copy/paste below while leaving out unnecessary
parts,
'Let $k^a$ represent the tangent vector field to a congruence of null
geodesics. We want to find a way to describe the behavior of a
neighboring collection of null geodesics in the congruence relative to
each other and for this we need a spatial deviation vector connecting
a given null geodesic in the congruence to infinitesimally nearby ones
in the congruence.' The equation turns out to be (also from the same
reference)
$$k^{c}\nabla_{c}\sigma_{ab} = -\theta {\sigma}_{ab}+
> C_{cbad}k^{c}k^{d} $$
where $\sigma_{ab}$ is the shear, and describes the deformation of a
ball into an ellipse for instance, and $\theta$ is the expansion
factor of the volume. C is the Weyl tensor. Those are in the subspace
orthogonal to $k_a$. If $\theta$ = 0, then the C term in the equation
above is the rate of change of the shear along the null geodesic. This
is described for instance in Wald.
It has some very useful properties to describe and analyze how a shape in spacetime changes along null geodesics, I.e., along gravitational waves.
It is useful in analyzing gravitational waves. One is the classification of Petrov types, with a general Weyl tensor asymptotically (as we go towrds infinity, ie, further from the course) able to be described as a sum of successively faster decaying Weyl Tensors, as higher inverse powers of u, with u a parameter along the null geodesics towards infinity. This is called the Peeling Theorem, and allows for the classification of Ricci flat spacetimes, and analysis of gravitational waves in vacuum. See a simple description at https://en.m.wikipedia.org/wiki/Peeling_theorem
Another useful tool derived from the Weyl tensor, the Newman-Penrose formulation, uses the Weyl scalars, 5 complex scalars which are formed from the 10 independent components of the Weyl tensor. One of those complex scalars is the outgoing gravitational wave, another is incoming, one is a static term (like the static Schwarzschild field), and two define gauges, at large distances (i.e., as u gets large). See https://en.m.wikipedia.org/wiki/Weyl_scalar
Best Answer
We will work with the metric supplied,
$$\mathrm{d}s^2 = A(r)^2\mathrm{d}t^2 -B(r)^2 \mathrm{d}r^2 -r^2 \mathrm{d}\theta^2 - r^2 \sin^2\theta \, \mathrm{d}\phi^2$$
I will assume that omitting the fourth spatial coordinate (which I have added) is simply a typo in the original post. I have also redefined the arbitrary functions for convenience. We choose a basis,
$$e^{t} = A(r)\mathrm{d}t, \, \, \, e^{r}=B(r)\mathrm{d}r, \, \, \, e^{\theta} = r\mathrm{d}\theta, \, \, \, e^{\phi} = r\sin \theta \, \mathrm{d}\phi$$
We now compute the exterior derivative of all $e^{a}$, and re-express them in terms of the basis,
$$\mathrm{d}e^{t} = -\frac{A'}{AB} e^{t} \wedge e^{r}, \, \, \, \mathrm{d}e^{r} = 0, \, \, \, \mathrm{d}e^{\theta}=-\frac{1}{rB} e^{\theta} \wedge e^{r}, \, \, \, \mathrm{d}e^{\phi} =-\frac{1}{rB}e^{\phi}\wedge e^{r} - \frac{\cot \theta}{r} e^{\phi}\wedge e^{\theta}$$
The spin connection $\omega^{a}_{b}$ can be read off from Cartan's first equation,
$$\mathrm{d}e^{a} + \omega^{a}_{b}\wedge e^{b} = 0$$
As the exterior derivatives are already in terms of the basis, this is fairly straightforward:
$$\omega^t_r = \frac{A'}{AB} e^{t},\, \, \, \omega^{\theta}_{r} = \frac{1}{rB} e^{\theta}, \, \, \, \omega^{\phi}_{r} = \frac{1}{rB}e^{\phi}, \, \, \, \omega^{\phi}_{\theta} = \frac{\cot \theta}{r}e^{\phi}$$
All other connection components presumably vanish. To compute the curvature 2-form, $R^{a}_{b}$, the second Cartan equation is required,
$$R^{a}_{b} = \mathrm{d}\omega^{a}_{b} + \omega^{a}_{c}\wedge \omega^{c}_{b}$$
An example of a particular component,
$$R^{t}_{r}= -\left( \frac{A''B -A'B'}{AB^3}\right) e^{t}\wedge e^{r}$$
Often $\omega^{a}_{c}\wedge \omega^{c}_{b}$ will simply be zero because of the few non-zero spin connections. After you have found all the components of the 2-form, the Riemann tensor is given by,
$$R^{a}_{b} = \frac{1}{2}R^{a}_{bcd} e^{c}\wedge e^{d}$$
For our example, this implies,
$$R^{t}_{r t r} = -2\left( \frac{A''B -A'B'}{AB^3}\right)$$
Now recall that thus far we have been working in a particular orthonormal basis. To find the Riemann tensor in the coordinate basis, we use the relation,
$$R^{\lambda}_{\mu \nu \sigma}=(e^{-1})^{\lambda}_{a} R^{a}_{bcd} e^{b}_{\mu}e^{c}_{\nu}e^{d}_{\sigma}$$
where the l.h.s the curvature is in the coordinate basis, and $e^{a}_\mu \mathrm{d}x^\mu = e^{a}$. As you are aware, the Ricci tensor $R_{\mu \nu} = R^{\lambda}_{\mu \lambda \nu}$, which you can then use to solve $R_{\mu \nu} = 0$ which should, hopefully, be straightforward given this is a homework problem.
If you are unfamiliar with the Cartan formalism, see the gravitational physics lectures at http://perimeterscholars.org/, at approximately lecture 3-4.