How does /etc/initramfs-tools/conf.d/resume work?

When there's a device referenced by UUID there then the system is waiting for that device before it can resume, because it needs the data stored on it. If the device doesn't exist then the system is going through every way of finding a device, and waiting, so as to be sure the device doesn't exist before it fails-over to a normal boot.

The link to swap is that the swap partition is used to store the memory contents in order to retrieve them and perform the resume operation.

You can check if swap is loaded with swapon --show and if swap is configured correctly by comparing blkid and cat /etc/fstab | grep swap.

Logical volumes (man lvm, sudo lvdisplay) can complicate the swap situation too.