The word "Leasing" in the context of a network probably has something to do with DHCP leasing. Dynamic Host Control Protocol is what is used in a fairly common and somewhat specific situation: You have a router that handles some small number of connections that are going to be given local IP addresses dynamically through DHCP. These putative lease-seeking devices have no permanent address but DO have the ability to adopt an address granted via DHCP transactions.
Your router's sub-net can only contain / manage a limited number of addresses as determined by the sub-net mask. For most home routers, your ISP grants 250 addresses and that becomes the maximum number of devices that can take out a lease. If you have 251 devices, "last one in" loses because there are no leases left. Commercial and business routers can have more or fewer addresses to manage depending on what "class" of router was purchased and how the network mask was set. While it isn't a requirement, lease sizes are often integer powers of two.
Using UDP (datagram protocol), devices on the sub-net request an address from the range "held" by the sub-net router. If that router has a free lease, it issues it to the requesting computer. That lease temporarily assigns a network IP address within the sub-net range of the router. Since routers perform Network Address Translation most of the time anyway, your computer's actual IP address almost doesn't matter because only the sub-net ever sees it. What matters is that through NAT, your session's traffic doesn't get mixed up with traffic from other sessions on the same router.
The concept of "leasing" in the context of database corruption comes up like this: To prevent "lease hoarding," leases have lifetimes - typically 24 hours but anything is possible from several hours to several days. At the end of the lease's lifetime, the lease and the IP address get reclaimed.
Here is the significant part. If the leased IP address was busy at the time, dropping the lease drops the connection. For Access databases, being on a leased IP that suddenly loses its lease is pretty catastrophic, and it is in that context that I think the earlier referenced article is talking about leasing. Disabling "leasing" means that once you have an address, you hold it until you actually break the session. It doesn't time out. This mode of operation on a DHCP sub-net avoids the ugliness of dropped network connections but exchanges that ugliness for the problem that if you actually lose your network connection by physical failure, the router won't get the signal to release the address. So they are subject to accidents that cause address losses until you restart the router.