Still don’t understand why a trade deficit must be financed by foreign borrowing (even after reading other answers)

The assertion that a trade deficit must be financed by foreign borrowing is technically incorrect (although it is often approximately true).

• The accounting identity for international transactions has a few extra terms, one of which is grants between governments (foreign aid). A country could run a trade deficit financed by foreign aid.
• Equity financing also shows up in international transactions. Instead of emitting debt, a country can emit equity.

The reason why the statement is approximately correct is that the largest new flow across borders is the transfer of debt instruments, noting that currency notes are a liability of government. Since net imports had to be paid for with some instruments (possibly bank deposits), there has to have been an offsetting flow of instruments out of the country. Since those instruments are debt instruments, people loosely refer to this as “borrowing,” although it could be the transfer of a bond issued years earlier. Most people do not refer to selling a bond out of their portfolio as “borrowing,” so one needs to be careful of this terminology.