A reciprocal lattice plane is a plane that passes through the reciprocal lattice points and whose successive stacking generates the entire reciprocal lattice.
Let the real space lattice translation vector be $\mathbf{T}$ which can be written as a linear combination of the primitive lattice vectors $\mathbf{a_1}$, $\mathbf{a_2}$, $\mathbf{a_3}$. Note that this linear combination is restricted in the sense that the coefficients can only be integers. Thus considering an arbitrary origin, the location of a point (similar to the position vector in classical mechanics) is given by $\mathbf{T}=p\mathbf{a_1}+q\mathbf{a_2}+r\mathbf{a_3}$ with $p,q,r \in Z$. You might think of reaching any point in the real space lattice by moving in multiples of only $\mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3}$. Note that you cannot reach any arbitrary point in space but can only reach discrete points. This is the essence of a lattice.
Now, the reciprocal lattice is constructed by the same mechanism, but with lattice vectors $\mathbf{b_1}$, $\mathbf{b_2}$, $\mathbf{b_3}$ with the condition that $\mathbf{a_i}.\mathbf{b_j}=2\pi \delta_{ij}$. Hence, the reciprocal lattice translation vector $\mathbf{G}=h\mathbf{b_1}+k\mathbf{b_2}+l\mathbf{b_3}$. By the same reasoning, the reciprocal lattice is again discrete. It is important to realise that the reciprocal lattice doesn't live in the same space as the real lattice.
Now, a real/reciprocal lattice plane is one which contains the real/reciprocal lattice points. When viewed from "above", that is, the direction of the normal, all the lattice planes corresponding to the same Miller indices look alike. Since the lattice is discrete and not continuous, there is a separation between the lattice planes. This can be calculated to be $$d_{hkl}=\frac{a}{\sqrt{h^2+k^2+l^2}}$$ Note that the above formula is valid only for cubic systems.