What stops messenger RNA from binding to itself?
It does fold on to itself. There are secondary structures in RNA and some of these secondary structures also have regulatory functions (for example, riboswitches). Some of these structures can also inhibit translation (by different mechanisms such as the masking of the ribosome binding site or analogous eukaryotic sequences, or stalling of ribosome etc). Other non-coding RNAs like tRNAs and rRNAs fold into specific structures that are essential for their function.

Many assays have been developed to study RNA secondary structure in vivo. SHAPE and DMS-seq are some examples.

The translation initiation factor eIF4A has a helicase activity that helps in resolving some secondary structures and facilitate translation (despite this, some structures can impede translation). 

Also check out ribosome profiling experiments.