Multi-AP Roaming Network Background
There is no magic to making multiple-AP (roaming) 802.11 networks work. Wireless clients just assume that all APs with the same SSID are configured similarly and are all just different points of access to the same underlying wired network. A client will scan all channels looking for APs publishing the SSID it wants, and will pick whichever one suits its needs best (usually that means whichever one shows the highest signal strength).
Once on the network, clients stay with the same AP as long as it is meeting the client's needs (i.e. as long as its signal strength is above a "good enough" threshold). If the client later thinks it could be better off with another AP on that network, it will do periodic scans of all channels looking for other APs publishing that SSID. If a scan turns up a candidate AP that is enough better than the AP it is currently on, it will automatically roam to the other AP, without dropping connections.
One roaming caveat: As another commenter pointed out, there are definitely poorly engineered clients out there with poor roaming algorithms or thresholds, which don't actually roam when they should, and thus end up being too "sticky", staying on the first AP they joined well after they could have been getting better performance and reliability with another AP that they are now closer to. Sometimes it helps to force the client's Wi-Fi interface to rejoin the network when you notice that a client has stuck to the wrong AP. If you have a lot of these buggy clients, then using the same SSID for multiple APs might not work well for you; you might want to use different SSIDs so you can more easily monitor and control which AP your client is associated to.*
Assuming both APs are configured similarly and are connected to the same underlying network, roaming is seamless and invisible to the user (except nerds like me who run tools to watch for these things). Roaming events are invisible to applications using the network, although some low-level parts of the network stack might be notified of the event, so that, for example, your DHCP client can double-check that this new AP really is connected to the same network, so it can be sure your DHCP lease is still valid on this network.
Some other users' Answers and Comments on this question erroneously suggested that wireless protocols or features like wireless relay or WDS might be needed for roaming, but that is absolutely incorrect. Those features are just ways to replace a wired Ethernet backhaul with a wireless one.
For the sake of completeness, I should mention that there have been a number of addenda to 802.11 over the years, such as 802.11F, k, r, u, and v, that have created mechanisms that allow coordinated multi-AP systems to help clients roam appropriately and quickly by doing things like keeping clients well-informed about their roaming options, and optimizing re-authentication when roaming, especially when tricky enterprise-class security is in use. But please note that none of those addenda are necessary for roaming to work. The roaming decisions are still ultimately up to the client, and the coordination among the APs does not require wireless relay, WDS, or mesh.
Configuration Suggestions
Give both APs the same network name (SSID), the same security type (WPA2/WPA3-PSK recommended; do not leave original WPA (TKIP) enabled or available in any way), and the same wireless security passphrase. Many clients assume that these kinds of settings will be the same across all APs with the same SSID.
Since you already have the cabling in place, use wired Ethernet as your backhaul. This saves your wireless bandwidth for your portable/mobile devices that actually need it, instead of wasting in on stationary devices like APs that could reasonably be cabled up.
If you have another device on the network, such as a broadband home gateway, providing NAT and DHCP service, then put both APs in bridge mode (turn off NAT and DHCP service). You generally only want one box on your network acting as a NAT gateway or serving DHCP. If you don't already have another device on your network doing NAT and DHCP, and you need those services, then you can have one of your APs do it. Have the more "upstream" AP (the one that's closer, topologically, to your broadband modem) do NAT and DHCP, and make sure that the wired Ethernet connection to the other AP comes from the first AP's LAN port. Also make sure that the "downstream" AP is in bridge mode. I call this out because I have seen people make the mistake of leaving NAT and DHCP enabled on both of their APs, and I have seen clients that are not smart enough to realize that, say, the 192.168.1.x/24 network they are on now is not the same 192.168.1.x/24 network they were on a moment ago in the other room. I have also seen users get confused in this situation where two laptops in the same house had 192.168.1.x addresses, but could not ping each other because they were really on two separate IP networks behind two separate NATs.
Channel is one key setting you do want to vary from AP to AP in a roaming (multiple AP) 802.11 network. To maximize bandwidth, leave your APs to automatically select the channel to use, or you can manually pick different, non-overlapping, and hopefully unoccupied channels to use. You don't want transmissions to/from one AP to compete for bandwidth with transmissions to/from the other AP.