In short this is the way I imagine them, as slight directional changes in spins propagating through the system.
The Heisenberg Hamiltonian for the exchange energy associated with magnetic coupling is a pairwise sum over an exchange integral $J_{ij}$ for two sites $i$ and $j$, with spin moments $\hat S_i$ and $\hat S_j$.
\begin{equation}
H_{ex}=-\sum _{i<j}J_{ij}\hat S_i\cdot \hat S_j
\end{equation}
Imagine a periodic array of lattice points each with a spin of two possible orientations $\pm\frac 12$, for a ferromagnetic ground state $(J_{ij}>0)$ with all the spins aligned, denoted as;
\begin{equation} |\uparrow \uparrow \uparrow \dots \uparrow\uparrow\rangle
\end{equation}
A low energy excitation would intuitively be a single flip of a spin somewhere in the lattice e.g.,
\begin{equation}
|\uparrow \downarrow\uparrow\dots\uparrow\uparrow\rangle
\end{equation}
However, this is not an eigenstate of the spin Hamiltonian. Instead low lying excitations have a complex wave character that is akin to a fractional displacement of a spin with respect to its neighbour. Such infinitesimal variation I personally imagine as a single reversed spin distributed over all the spins in the lattice.
Therefore you could view a spin wave as a superposition where all the spins contain a bit of this single inverted spin in the first excited state. It is easy to see that this is the case since we are using ladder operators to change our spins from up to down. Acting as a superposition over the lattice (spin wave) reduces the disturbance between any given adjacent pair, whereas the naive expected eigenstate would include a really big difference in exchange energies at a localised region.
As you may be thinking, how can I have an infinitesimally displaced spin when there are only two spin orientations? Well that is a bit of a misnomer as this oversimplified model has a discrete symmetry of 2. In reality there is a smooth symmetry associated with these spins so that the spin wave is actually a more realistic picture than the 2-fold orientation (Ising model).
Hope that helps :)
