Insulin, like glucagon, binds to receptors on the surface of muscle cells and illicits a response that is transduced to produce changes of phosphorylation in the enzymes controlling glycogen turnover. Unlike glucagon it does not initiate an enzyme phosphorylation cascade, but its binding to the receptor causes autophosphorylation on tyrosine residues, resulting in protein–protein interactions which lead to the activation of various enzymes through subsequent phosphorylations.
I would recommend you go to Berg et al. in NCBI Bookshelf online — rather than Wikipedia — for this kind of thing. It is a best-selling biochemistry textbook which is produced with great care (even if the edition online is out of date). Thus in Section 21.5.2 you will find:
How is glycogen synthesis stimulated? As stated earlier, the presence of glucagon signifies the starved state and initiates glycogen breakdown while inhibiting glycogen synthesis. When blood-glucose levels are high, insulin stimulates the synthesis of glycogen by triggering a pathway that activates protein phosphatase 1 (Figure 21.20). The first step in the action of insulin is its binding to a receptor tyrosine kinase in the plasma membrane. Multiple phosphorylations again serve as the instigation for a regulatory wave of dephosphorylations. The binding of insulin to its receptor leads to the activation of an insulin-sensitive protein kinase that phosphorylates the RGl subunit of PP1 at a site different from that modified by protein kinase A. This phosphorylation leads to the association of the RGl subunit with PP1 and the glycogen molecule. The consequent dephosphorylation of glycogen synthase, phosphorylase kinase, and phosphorylase promotes glycogen synthesis and blocks its degradation.
I am assuming that you are familiar with the pathways of glycogen synthesis and breakdown. (If not, search for this in Berg et al.) At the risk of violating copyright, I reproduce Figure 21.20 below:
You can read more about the insulin receptor in Section 15.4.1 of Berg et al.
Insulin also affects glycogen synthesis indirectly by stimulating/activating the glucose transporter, GLUT4. The mechanism by which this is accomplished is less well established than that described above.