Why does the balance sheet balance?

The balance sheet balances because that is how it is defined. By definition,

Assets - Liabilities = Shareholders' Equity

Hence on the balance sheet, Assets are shown on the left, Liabilities and Shareholders' Equity are shown on the right. To illustrate, assuming a start-up company borrowed $100 to buy a printer that cost $200 with initial capital of $100. Its balance sheet would look like this:

Assets (printer) $200
Liabilities $100
Equity (or initial capital) $100
Assets $200 - Liabilities $100 = Shareholders' Equity $100

This has to be the case because initial capital and borrowing add up to the value of the asset that the company has for it to run its business. Now assume that after 1 year, the company generates $20 of profits by printing and selling digital pictures. Its balance sheet would look like this:

Assets $220 (Printer $200 + Cash from profits $20)
Liabilities $200
Equity $120
Assets $220 - Liabilities $100 = Shareholders' Equity $120

Again the balance sheet balances because any entry into any parts of the balance sheet would have a corresponding cancelling entry on another part. In this case, Assets increases by $20 and Equity by $20 as well.