I understand this to be more of a legal question than historical, though the two are interconnected.

Canada's Head of State is currently Queen Elizabeth II. As Canada's governmental structure is founded on the British Westminster system, the Queen retains powers, rights and vestments that are not conferred onto the Government of Canada. One of these vestments is Head of the Armed Forces. At the time of Confederation, the British North America Act (or Constitution Act, 1867) ensured that vestment continued with the Queen:

Section 15: "The Command-in-Chief of the Land and Naval Militia, and of all Naval and Military Forces, of and in Canada, is hereby declared to continue and be vested in the Queen."

As you likely know, the Governor General is the Queen's representative in Canada. This means that he (the current Governor General is David Johnston) wields those powers conferred to him by the Queen, at her pleasure. In 1947, The Letters Patent Constituting the Office of the Governor General constituted, as the name suggests, the Office of Governor General and Commander-in-Chief, which was to be filled by one person, the Governor General. Article IX of those Letters Patent require and commands all Officers and Ministers, Civil and Military, ...to be obedient, aiding, and assisting unto the Governor General.

This is how the Governor General has the title of Commander and Chief of the Canadian armed forces.

Why not vest it in the Prime Minister? He is not the Head of State. He is only the Head of Government. This is my first post, so I will try to adhere to the subreddit policies about digressing. A good followup question would be to ask about the role of the sovereign in Canadian legal history.

I should also note that you a very liberal with the term 'symbolic'. You are quite right that the modern Governor General has a largely symbolic role, however, this is a matter of tradition, not law. The Governor General wields real power and can, for example, refuse the advice of the Prime Minister as was seen in the King-Byng Affair and more recently in the prorogation crisis of 2008.

edit: removed a redundant sentence and cleaned up the final paragraph. edit2: more clean up