The short answer is that we won because the Americans launched an offensive war aimed at specifically changing the status quo relationship between them and Britain in North America, and did not accomplish that. If you buy that: since the US lost, we in turn must have won. I'm probably more willing to accept the argument that while no one really won the War of 1812, it's pretty clear that Indigenous people lost it.

The first idea is pretty straight forward: America went to war to change their relationship with Britain. You will see a lot of people talk about causes of the war like impressment, illegal blockades, control of the seas, and British Agents interfering with Indigenous peoples on America's western frontier, but I don't think these adequately explain why war broke out in 1812. Although they are listed in Madison's war message, these had been ongoing issues between Britain and America - so why declare war in 1812, and not earlier? Why try to suddenly occupy British colonies in North America? I am persuaded by JCA Stagg, who writes in Mr. Madison's War that

with a total population of barely half a million, the various provinces of Canad seemed to be the weakest links in the chain of British imperial power and many Americans assumed that they could be easily seized by the United States with its vastly superior population of nearly seven and one half million. Yet Canada did not seem in itself to be the source of the most important grievances that the United States wish to settle by war... For this reason, most opponents of the war never ceased to point out that the conquest of Canada promised neither to guarantee respect for American maritime rights nor even to reimburse the nation for the expense of the effort.

Stagg argues effectively that Madison was trying to force Britain to accept American proposals concerning those original issues that deal with the seas and the Western Frontier by depriving them of their commercial base in Canada. Then Britain would have no alternative to American markets and economic influence. So it IS a war over territorial control, not, as you might think, simply about "non Canadian" concerns.

This is all well and good, but Madison had little reason to believe that this would work other than from his own idealism. The British strength at sea was absolute, far out matching the Americans, nor did the Americans have any real strategic plan to take Canadian strong points like Quebec or Montreal. He had ambiguous strategic objectives matched with an unclear operational focus - a poor start for any war.

As a result of this and due to some other internal factors, the Americans launched a terrible war effort. Instead of going for the chokepoints at Montreal or Quebec, they drove up through Ontario after repelling the initial British foray into Ohio via Detroit. They then spent years successfully forcing the British (who were occupied in Europe) back towards York (present day Toronto), and achieving dominance over the Great Lakes - but to what end? Britain could still send in troops through the St Lawrence and Quebec all the way to Kingston. On the high seas of the Atlantic, the Americans were beaten time and again. The British sailed in and attacked Washington with little opposition. In the west, British/allied forces filtered down through Wisconsin, Illinois and upper Michigan. Any American victories did little to achieve their own aims, while a longer war only helped Britain.

When the war finally ended, the Americans were in a bad long term position, but a good short term one. The December 1814 Treaty of Ghent was signed as news of British defeats at Baltimore and Lake Champlain arrived, and as European instability hemmed and hawed while Napoleon was in exile. The result was the Americans successfully pressed for a status quo peace and the British, who were more concerned about Europe, accepted it. However, before those 1814 defeats, Britain had been pressing for sweeping changes that would have given parts of New York and Maine to Canada, and ensured the protectiion of the western "Indian Territory." The US was in no position to negotiate by war's end, and I'd say they were lucky to get a status quo peace. If British troops in Europe had been able to sail west after defeating Napoleon's 100 Days and naval operations continued into 1815-16, the Americans would have little chance against the might of the British Empire.

Another unfortunate result is my second point about Indigenous people losing the war. While status quo was maintained generally, the British lost interest in protecting the "Indian Territory." The Americans - out of a desire to deal with a rapidly growing population and perhaps out of revenge for many Indigenous groups' allegiance to Britain - drove westward and began a colonization of the midwest that forced them out of their homes and further and further into the plains. The assertion of American sovereignty across the continent began in the aftermath of 1812 and would have dire consequences for North American indigenous peoples.

I guess, TLDR: Canadians think they won because they were on the winning side.