In the most plain terms, most assets are priced in dollars. $60/share means that each share costs 60 dollars. It's a coefficient in the numerator of an equation.
(60 * dollar)/ 1 share = stock's value
If "stock's value" is held constant, then a decrease in the dollar means you have to have a compensating increase in the number (60) to balance the equation.
Of course, it's more complicated than that in reality. A decline in the dollar is generally seen, these days, as an increase in liquidity, which drives up asset prices. A strong dollar should be good for stocks if we emerge from the liquidity/credit collapse phase. But it may be a problem for companies who make most of their money in foreign markets b/c they have to exchange revs back into dollars to run operations. This exchange is more costly when the dollar is rising. For small caps, it generally helps because most of them run ops here, and a strengthening dollar increases their purchasing power for imported resources.
I think that about skims the surface. But it's of course even way more complicated than that and all you can really do is be aware of which aspect of the relationship the market seems to be responding to during a given period.
Right now, markets rally on weak dollar moves, and come in on dollar rallies. That would probably change if some sort of dollar panic took hold.