What's a template switching site?
From the page you cited:


  During first-strand synthesis, upon reaching the 5’ end of the RNA template, the terminal transferase activity of the MMLV reverse transcriptase adds a few additional nucleotide


That usually are CCC (1st is almost always C and the other two most of the times), see: Zajac et al., 2013

These CCCs are used for priming the Template Switching Oligo (TSO), that has GGG and whatever you want to add to the sequence (for instance UMIs and adapters). 

So, answering to your question the polimerase does not switch strands, it switches from mRNA as a template to TSO as template. Because of that is called template switching.



You can see it in the graph from the page you cited. In the template switching step polimerase switch from the mRNA to the TSO and continue with the synthesis of the first cDNA strand using TSO as template.