All other things being equal, a weak dollar will lead to an increase in exports to countries with currencies that have grown stronger relative to the dollar. We must remember that a dollar that is weak against one currency may be strong against another so we must not assume that exports to all countries will be enhanced.
The reason that exports will increase is that the people in the foreign countries who buy American exports need to pay for them in dollars. What typically happens is that the foreign buyer uses their own currency to buy American dollars which are then used to pay the American exporter. If the dollar becomes weaker, the other currency becomes stronger. When that happens, the foreign buyers do not have to use as much of their money to get a certain amount of dollars (and, thereby, a certain quantity of goods or services). In essence, a weaker dollar reduces the prices of American exports from the point of view of the foreign buyer.
For this reason, a weak dollar will (again, all other things being equal) lead to an increase in American exports.