foreign borrowing for stimulus

As I understand it, recessions are triggered because people hold onto
money they would otherwise spend. And the idea of stimulus is to
borrow a good part of that money people are holding onto, and spend it
for them. This gets the economy going again, causing people to regain
confidence, and stop trying to hold onto so much money.

We keep hearing that the stimulus bill recently passed by the US
Congress relies on foreign borrowing to pay for it. But why can't the
US government borrow instead from the extra money that Americans are
holding onto? Are Americans putting their savings into foreign banks,
or into their mattresses? Is it because the level of demand prior to
the recession relied on foreign borrowing to finance it, so the
government needs foreign borrowing to recreate the same level of
demand? And if so, isn't this unsustainable, doesn't the US just need
to create a safety net for the hardest hit, and otherwise just take
its licks?