I'm trying to understand how MikroTik works in regards to dual-SIM setsup. For example, the LtAP has 3 SIM slots, but only one modem.

I'd assume that the GUI has a way to select which SIM to use?

In terms of automatic failover, is scripting typically used to check if the current SIM's carrier is connected, and if it's not, to switch to a different one?

You can add move LTE Modem with PCI if you want to use all of 3 SIM Slot.

For detail about how to configure automatic SIM Fail Over you can follow this link : https://wiki.mikrotik.com/wiki/Dual_SIM_Application.