The tax brackets are for amounts of taxable income. The personal allowances specify the amount of income that is not considered taxable.
If you earn $15,100 then your first $15,000 is not taxable. You have a taxable income of $100, and you pay 15% ($15) in tax on it, since it all falls in the first bracket.
If you earned $65,000 you would have a taxable income of $50,000 and pay $7,500.
What may be throwing you off is that some tax calculators include EI and CPP contributions as taxes. Those contributions don't follow the brackets, so you can be below the lowest tax bracket threshold and still be charged "tax" (which is really EI and CPP).