Use cases of RNA secondary structure prediction
Short answer

I can think of at least a dozen applications for which it would be useful to know the secondary structure of a given sequence of RNA off of the top of my head. In no particular order:


Simulation/visualization of RNA
Riboswitches
MicroRNA
RNA interference (RNAi)
RNA-RNA interactions
RNA-DNA interactions
RNA-protein interactions
Ribosomal protein expression
Ribozymes
Forced evolution of RNA aptamers
Synthetic tRNAs with four and five base-pair codons
Termination of transcription (the process that makes mRNA from DNA)


Basically anything to do with RNA, which is one of the fundamental building blocks of all living creatures.

More in-depth

Simulation/visualization of RNA

Secondary structure (2°) forms the second (yep) level in the standard 4 level hierarchy of macromolecular structure, and consists of short-range interactions between and among residues. In RNA, 2° is usually thought of as being loosely identical with base-pairing interactions. In order to do a proper simulation/visualization you'll probably also need at least the 3° information, but 2° is important too.

Riboswitches

As an mRNA gets translated by a ribosome, in many cases it can form into a complex 3D structure that will affect the translation process. For example, some of the mRNAs that encode proteins related to the metabolism of metal ions have a riboswitch that will themselves bind to metal ions and inhibit/upregulate expression of the related protein. For various reasons it's usually extremely difficult to get the full 3D structure of a riboswitch, so getting to know the 2° remains an important source of information about them.