What is the biological significance of k-mer counting?
The applications for counting the k-mer occurences in a sequence are: building de Bruijn graphs [1] for de novo assembly from very large number of short reads, produced by next generation sequencing, fast multiple sequence alignment [2], and [3] repeat detection. 


  
  Compeau PE, Pevzner PA, Tesler G: How to apply de Bruijn graphs to genome assembly. Nature Biotechnol. 2011, 29 (11): 987-991.
  10.1038/nbt.2023
  Edgar RC: MUSCLE: multiple sequence alignment with high accuracy and high throughput. Nucleic Acids Res. 2004, 32 (5): 1792-1797.
  10.1093/nar/gkh340
  Kurtz S, Narechania A, Stein J, Ware D: A new method to compute K-mer frequencies and its application to annotate large repetitive
  plant genomes. BMC Genomics. 2008, 9 (1): 517-10.1186/1471-2164-9-517
  


To be clear, here is the biological significance of each of these applications: 


Sequencing any genomes (bacterial, plant, mammalian) has become very cheap and easy recently with the next generation sequencing technologies. As opposed to the Sanger sequencing (the first generation), these produce as a readout very short sequencing reads, which have to be assembled into longer ones, and subsequently, into whole genomes. Counting the k-mers is part of the assembly process and this is its biological importance.
In order to compare whether proteins might have similar function, and thus belong to the same protein family, a multiple sequence alignment is performed. This is crucial step when annotating new protein sequences and studying their biological functions.
There are many repeats in our genome, as well as in plants. Repeated sequences are also involved in neurodegenerative diseases such as Hungtingtons, so being able to identify them is of medicinal importance.