Wireless internet signal bi-directional amplification?
Regarding the first part of your question, as to how a bad signal causes slower speeds:
The TCP protocol used for most internet traffic has error detection. Basically, each packet has a checksum so the receiver can know whether it is correct. The receiver then sends an acknowledgement back to the transmitter that the packet was received. The transmitter will stop and re-send any packets that are not received, possibly more than once. This can significantly lower communication speed. See the Reliable Transmission section on the TCP Wikipedia page for more information.
As to your second question about whether this will work: In theory, yes, such a system can work. However, in practice, neither you nor I have enough knowledge or experience to do so correctly. Fortunately, WiFi extenders are available for sale, so you may be able to use those. I've used them before myself, but kept having to restart them and disconnect/reconnect so often (maybe hourly) that I found them very frustrating. Probably if you spend enough money you can get a reliable one.
What I ended up doing in my house, since the modem connection from outside was at a far corner, was running an ethernet cable from the modem through the walls into a closet located near the middle of the house, and connecting a dedicated WiFi router from there so the entire house had a good signal.