How does the government collect GST?

In Canada it is the registrants who collect GST/HST from the people to whom they provide their goods and services. Depending on their filing frequency of thier GST/HST returns (monthly, quarterly or annually), they indicate the amount of GST/HST that they have collected as well as the GST/HST that they paid as inputs to their commercial activity (input tax credits). The input tax credits are deducted from the GST/HST collected and the balance (positive or negative) becomes their net tax. They then either remit (if a positive amount) or claim their "refund" (negative amount).

The government doesn't actually "collect" the tax, they receive it from their tax collectors...the GST/HST registrants.