I don't think it really explains what it's for though. Here's my explanation (from my PhD thesis on compilers):
Static single assignment (SSA) form provides an efficient intermediate representation for program analysis [Cytron et al., 1989, 1991]. It was introduced as a way of efficiently representing dataflow analyses.
SSA uses a single key idea: that all variables in the program are renamed so that each variable is assigned a value at a single unique statement in the program. From this key idea, SSA is able to provide a number of advantages over other techniques of dataflow analyses:
Factored use-def chain: With a def-use chain, dataflow results may be propagated directly from the assignment of a variable to all of its uses. However, a def-use chain requires an edge from each definition to each use, which may be expensive when a program has many definitions and uses of the same variable. In practice, this occurs in the presence of switch-statements. SSA factors the def-use chain over a φ-node, avoiding this pathological case.
Flow-sensitivity: A flow-insensitive algorithm performed on an SSA form is much more precise than if SSA form were not used. The flow-insensitive problem of multiple definitions to the same variable is solved by the single assignment property. The allows a flow-insensitive algorithm to approach the precision of a flow-sensitive algorithm.
Memory usage: Without SSA form, an analysis must store information for every variable at every program point. SSA form allows a sparse analysis, where an analysis must store information only for every assignment in the program. With a unique version per assignment, the memory usage of storing the results of an analysis can be considerably lower than using bit-vector or set-based approaches.
Static single assignment (SSA) form provides an efficient intermediate representation for program analysis [Cytron et al., 1989, 1991]. It was introduced as a way of efficiently representing dataflow analyses.
SSA uses a single key idea: that all variables in the program are renamed so that each variable is assigned a value at a single unique statement in the program. From this key idea, SSA is able to provide a number of advantages over other techniques of dataflow analyses:
Factored use-def chain: With a def-use chain, dataflow results may be propagated directly from the assignment of a variable to all of its uses. However, a def-use chain requires an edge from each definition to each use, which may be expensive when a program has many definitions and uses of the same variable. In practice, this occurs in the presence of switch-statements. SSA factors the def-use chain over a φ-node, avoiding this pathological case.
Flow-sensitivity: A flow-insensitive algorithm performed on an SSA form is much more precise than if SSA form were not used. The flow-insensitive problem of multiple definitions to the same variable is solved by the single assignment property. The allows a flow-insensitive algorithm to approach the precision of a flow-sensitive algorithm.
Memory usage: Without SSA form, an analysis must store information for every variable at every program point. SSA form allows a sparse analysis, where an analysis must store information only for every assignment in the program. With a unique version per assignment, the memory usage of storing the results of an analysis can be considerably lower than using bit-vector or set-based approaches.