It means that officially, she is the Head of State. Being Head of State means you're the official representative of the country in public affairs. This is different from being Head of Government, which means you're in charge of running the country: in the Commonwealth countries, the HoG is the Prime Minister.

Theoretically, the Queen does have powers to veto laws, dissolve the legislature, etc. (which is why laws are still given the "Royal Assent" when they are passed, and why you always hear things about the Prime Minister asking the queen to dissolve parliament). In practice, the queen devolves these powers to a representative (The Governor-General), who carries out these powers on her behalf. Furthermore, while theoretically, she has a lot of power, in practice, none of this is ever used except at the request of the ruling government.

The Queen's Head of Stateship derives from histroical ties; when the Commonwealth countries gained independence from Parliament in Westminster, the colonies decided to continue accepting the Queen as head of state. Think of the queen's rule this way: she is Queen of the United Kingdom, AND of Canada, AND of Australia AND of New zealand etc. She is not queen of the United Kingdom and THEREFORE Queen of Canada. She is simply queen of multiple countries at once, all of which have broadly the same succession laws (whenever the succession laws are changed, the British government tends to consult with the Commonwealth countries to ensure that these are maintained). This was particularly important during the Edward VIII abdication crisis, where the Commonwealth countries all individually had to agree to his abdication (in practice nobody liked the man, so this was done very quickly), and during the recent changes to the succession law to change to absolute primogeniture (i.e. firstborn child is heir to the throne) rather than male-preference primogeniture (i.e. a male will always inherit before his sister).

The independence of these countries was a gradual process, and not all achieved it at the same time. Most of these went through a phase where they were granted self-governing legislatures ("Responsible Government"), awarded formal dominion status, where a colony was self-governing but still subject to vetoes from London (Canada was first, in the middle of the 19th century), and then, after the Statute of Westminster in 1931 became Commonwealth Realms, made them de facto independent from the British state (i.e. legislative oversight from London).