Recently I heard it mentioned that the activities of the Fed (e.g.,
monetary policy) can affect the international exchange rate of the
dollar.

Your task is to spell out exactly how this is possible (assuming this
assertion is correct).

I understand the Fed and its three tools of monetary policy fairly
well, I think.  But while I understand how the monkeyings of the Fed
can increase or lower the interest rate, inflation, unemployment, and
the money supply . . . I don't necessarily see how any of this would
have any direct bearing on the international value of the dollar.

The Fed's monetary policy directly affects the forex price of the
dollar by altering the supply/demand balance in forex markets. When
the Fed expands the money supply (loose/expansionary monetary policy),
that increases the supply of dollars. All else equal, supply goes up,
price goes down. The opposite happens under tight/contractionary
policy.

Indirectly, the Fed acts as a barometer on the US economy. If the
Fed's actions are seen as indicitive of a healthy US economy, foreign
investors are more likely to want to invest here. This increases the
demand for dollars on the forex market, so price rises. The opposite
happens if foreign investors perceive the Fed's actions as indicitive
of a weakening economy.

Milton Friedman & Anna Jacobson Schwartz, "Monetary History of the
United States, 1867-1960"
http://www.amazon.com/exec/obidos/tg/detail/-/0691003548/qid=1117383052/sr=8-5/ref=sr_8_xs_ap_i5_xgl14/102-0226167-4750575?v=glance&s=books&n=507846

This is a very complex problem and it is called transmission mechanism
or transmission of the monetary policy. There are at least 4
traditional transmission channels in the economy: (1) credit channel,
(2) Tobin's Q channel,(3) interest rate channel and (4) households
expenditures channel.
However, they do not work any longer and therefore, Central Banks have
their analysts who try to estimate new models.
You can find traditional channels in Mischkin "Modern Banking" or any
other book about monetary policy. Bear that in mind, however, that
there are many contemporary models as we try to re-establish these
channels.