So this makes the router sends a stronger signal, farther. But how does this help if my laptop still has the same wifi adapter and can't send a strong signal back?
It also makes the antenna more sensitive in the direction its pointing. The elements make the antenna more directional, sort of focusing the energy. This increases the transmit side by concentrating the energy into a smaller area, but the same effect also helps it collect more energy.
The trade-off is worse performance perpendicular to the extra elements.
Antennas are coupled. Transmitting is receiving, and receiving is transmitting.
When you pull on a chain or a rope, it doesn't matter if the other side is "receiving" the pull, or "sending" it. The two sides are physically coupled.
Similarly, antennas are physically coupled to each other. Increase the gain on one side, the other side still benefits. You're physically connecting a "rope" of electrons over the air.
The trade-off is worse performance perpendicular to the extra elements.