whatever my laptop sends to my router will be what the router uses to match against the store value and grant access.
You seem to assume that the "Pre Shared Key" is used like a password that needs to be authenticated. It's not used that way.
The security key, since it is already installed on both the host (wireless router) and client (the laptop), is never transmitted over the radio. The security key is used to encrypt the data packet prior to transmision. The receiving end uses its copy of the security key to decrypt the radio message. That's why it's called a "key" (as in lock) rather than a "password".
There are communication situations where one end has the key and the other end does not. Various schemes (e.g. WPS) have been devised to send the key to the other end so that secure communication can commence. PC wireless networks normally avoid this conundrum (or security weak-spot) by requiring the security keys to be manually installed prior to making any connection.
So what stops someone just sniffing whatever I sent to the router and sending this to the router to gain access?
Any decent security protocol should have protection against message-capture-and-replay (aka "man in the middle attack"). Typically each message includes time-sensitive information, such as a message ID number or a timestamp, to make every message unique. A repeated or stale message would be discarded.