GDP Identity:Why does $text{C+S+T = C+I+G+(X-M)}$?

A concrete example makes it easier to think about:

Let's say during the week you receive $1000 income. Only 3 things can happen to your $1000:

1. You could be taxed some/all of it (T: Tax)
2. You could spend some/all of it (C: Consumption)
3. You could save some/all of it (S: Savings)

In this example we've used your $1000 income. The distribution of income (Y = C + S + T) is exactly the same but not just one week of your income, but 52 weeks of everyone's income. So it's a measure of how much income was received across everyone in the economy for the whole year.

Now, all this income had to originate from somewhere, and there are only four possibilities:

1. someone's spending (C: Consumption) - this goes to businesses and is paid out as wages, as interest on business loans, or as share dividends or profits to owners
2. government spending (G: Government Spending) - payments to government workers and for anything government buys (e.g. materials to build roads, defence spending etc etc)
3. banks making loans (I: Investment) - loans made to businesses or individuals
4. sales to foreign economies less what is paid to foreign economies for their goods and services (X - M: exports minus imports)

Adding these components together gives aggregate demand: C + I + G + (X - M); it's the $ amount of all the goods and services purchased in an economy over one year.

GDP is not a perfect way to measure the size of an economy, but it's probably the most common way. So when we talk about GDP = Y = C + S + T = C + I + G + (X - M), what we know is that GDP can be measured in a few different ways, and that more spending will mean more income (and vice-versa).