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I have extremely little knowledge about finance in general, but I have had a TFSA for several years now and know that my money keeps magically growing.
Exactly where does the interest I gain come from? And if it's tax-free, how is this "profitable" for the bank/government?
A TFSA is a tax free savings account. It is a type of account where you can buy various investments like stocks, bonds, or funds (mutual, exchange traded, and money market). There are some other options but it's best to see what your bank or broker will allow.
You probably specified the type of investment when you opened the account. You can look at your statements or maybe online to see what you're invested in. My guess is some kind of HISA (high interest savings account). This is kind of the default option for banks.
The government created these accounts for a variety of reasons. The main stated reason was to encourage people to save.
Obviously they also do things to get votes. There was an outcry after the change to a type of investment called "investment trusts". This could be seen as a consolation prize. These can be valuable to seniors for many reasons and they tend to vote more often. There was also an election promise to eliminate capital gains taxes in some fashion.
It's not profitable for the government, in fact it supposedly cost the federal government $410 million in 2013. Banks make money by investing your deposit or by charging fees.
You can see what every tax break 'costs' the government in lost revenue here http://www.fin.gc.ca/taxexp-depfisc/2013/taxexp1301-eng.asp#toc7
As to where the interest comes from: The same place it comes from in other kinds of savings accounts. The bank takes the money you deposit and invests it elsewhere, traditionally by lending it out to others (hence the concept of a "savings and loan" bank). They make a profit as long as the interest they give for "borrowing" from you, plus the cost of administering the savings accounts and loans, is less than the interest they charge for lending to others.
No, they don't have to pay you interest -- but if they didn't, you'd be likely to deposit your funds at another bank which did. Their ideal goal is to pay as little as possible without losing depositors, while charging as much as possible without losing borrowers. (yeah, I know, typo corrected)
Why do they get higher interest rate than they pay you? Mostly because your deposits and interest are essentially guaranteed, whereas the folks they're lending to may be late paying or default on those loans. As with any kind of investment, higher return requires more work and/or higher risk, plus (ususally) larger reserves so you can afford to ride out any losses that do occur.
