In 1962, Canada's immigration rules changed to a neutral, meritocratic system, to allow anyone to be admitted into the country regardless of race or nationality. Why did they do this? (more within)

The transition from the previously exclusionary immigration policy to the more meritocratic "points system" was made by the Immigration Act of 1967, passed by the Liberal government of Lester B. Pearson. The changes made in this act were based on a White Paper (i.e. government discussion paper) that was produced the previous year, which recommended greater clarity in Canada's immigration policy and a removal of the racially discriminatory aspects of the existing policy. Since Canada had signed the UN Charter in 1944, which included clauses that prohibited racial discrimination in immigration policy, policymakers had had to revert to coded language. The immigration policy prior to 1967 included provisions that immigrants had to be suited to the Canadian climate and emphasized the ability to assimilate into Canadian society. By 1967, the Canadian government considered these caveats to be a problem. Meanwhile, newly independent states in Africa and Asia had been raising the issue of discriminatory immigration policies at the UN, meaning that the issue was in the international zeitgeist.

There was also, as you suggest, a need to expand the immigrant pool by the 1960s. As early as the 1950s, immigration from northwestern Europe to Canada was declining. In the postwar decades the Canadian economy was booming, and labour shortages were starting to become an issue. The government began to recruit Italian, Portuguese, and Greek immigrants to fill these gaps, communities which soon became "the engine of the rapid urbanization of southern Ontario" (Mackey, 66). Even still, the rise in immigrants from southern Europe caused racial anxiety in Canada. When Italian immigrants surpassed British immigrants, one of Pearson's immigration ministers toured northern Europe in 1964 to try and renew interest in migration to "balance" the flow from other, less desirable countries. As labour shortages continued, however, more changes were necessary. The "points system" introduced in 1967 assigned points to different skills, education level, and other factors that would allow immigrants to make a contribution to Canadian society. Skilled immigrants from Asia and Africa now had a viable path to Canadian residence, and by the early 1970s immigrants from these nations had surpassed those from Europe.

These changes also occurred within a context of the "new nationalism" that began to take shape in English Canada during the 1960s and 1970s. Whereas English Canada had relied on British symbols and heritage for their national identity, the Pearson government oversaw a transition to a more distinctly Canadian national identity, one which sought to be more inclusive of French Canadians and other minorities. This move towards a sense of civic nationalism in Canada coincided (and competed) with the nationalist awakening in Quebec known as the Quiet Revolution. The new immigration policy that was adopted in 1967 was a part of this evolution, which also included things like the adoption of the maple leaf flag in 1965 and the push for official bilingualism. This process would continue under Pierre Trudeau, who adopted the Official Languages Act in 1969 and official multiculturalism in 1971. Opening the doors to non-European and non-white migrants was therefore part of the redefinition of Canada as its own nation, with its own flag, bilingual institutions, and multicultural heritage.

TL;DR: A need for immigrant labour to fuel the postwar economic boom in Canada coincided with a national identity in transition to force the government to re-examine its immigration policy.