ARP spoofing usually works by fooling all the clients into thinking that you're the router, by faking the ARP responses that translate IP addresses to MAC addresses. When clients receive the ARP response, they remember the MAC that was associated with the IP.
Once you stop the application that's handling the man-in-the-middle part of the operation, the clients keep sending to your MAC address, instead of the router's. Since you're no longer handling such packets, the traffic is blackholed and the whole network goes down. Resetting the router causes it to send an ARP broadcast (e.g. "Hi, I'm
12:34:56:78:90:AB") along with a DHCP broadcast, allowing clients to re-sync with the real router.
It may be possible for your ARP poisoning software to send out an ARP broadcast when it closes, with the real MAC address of the router, in order to prevent this. This may be a bug, or it may just not be implemented yet.