Usually a bug is exploited to temporarily get a chosen process running as root, that is used to remount the file system read write, either startup scripts are changed and/or suid binaries added, then the file systems are synced and remounted read only again and likely the device is rebooted.
You don't normally want to have the system partition writable on an android device, even when rooted - you only make it writable on the rare occasions when you are changing it.
EDIT-Summary of comments: It appears that /system on this device is cramfs, an inherently read-only compressed file system. Likely a modified image would need to be generated on a PC and then flashed.
Search result for dealing with this problem on rockchip based devices: