In case you will accept a concise answer...
What questions does it answer? Visual mapping of pairwise dissimilarities in euclidean (mostly) space of low dimensionality.
Which researchers are often interested in using it? Everyone who aims either to display clusters of points or to get some insight of possible latent dimensions along which points differentiate. Or who just wants to turn a proximity matrix into points X variables data.
Are there other statistical techniques which perform similar functions? PCA (linear, nonlinear), Correspondence analysis, Multidimensional unfolding (a version of MDS for rectangular matrices). They are related in different ways to MDS but are rarely seen as substitutes of it. (Linear PCA and CA are closely related linear algebra space-reducing operations on square and rectangular matrices, respectively. MDS and MDU are similar iterative generally nonlinear space-fitting algorithms on square and rectangular matrices, respectively.)
What theory is developed around it? Matrix of observed dissimilarities $S$ is transformed into disparities $T$ in such a way as to minimize error $E$ of mapping the disparities by means of euclidean distances $D$ in $m$-dimensional space: $S \rightarrow T =^m D+E$. The transformation could be requested linear (metric MDS) or monotonic (non-metric MDS). $E$ could be absolute error or squared error or other stress function. You can obtain a map for a single matrix $S$ (classic or simple MDS) or a map for many matrices at once with additional map of weights (individual differences or weighted MDS). There are as well other forms like repeated MDS and generalized MDS. So, MDS is a diverse technique.
How does "MDS" relates to "SSA"? Notion about this can be found on Wikipedia page of MDS.
Update for the last point. This technote from SPSS leaves impression that SSA is a case of Multidimensional unfolding (PREFSCAL procedure in SPSS). The latter, as I've noted above, is MDS algo applied to rectangular (rather than square symmetric) matrices.