(NB: Tapping “learn more about trusted certification” doesn’t provide much of an answer, or at least, not for someone from my limited perspective.)
Others have given the correct answers but long story short, your phone (or your PC, tablet, etc) all have a list of common trusted "root" certification authorities. Verisign, Digicert, and many others are in there. Asus is not one of them. By toggling it on, you are simply preventing the warning of "this site is not secure" when you visit the page via HTTPS, at least when that error is due to the fact that router.asus.com is not a known authority.
It does not hurt anything to trust the cert, since you know the device and know it is not malicious. The "root" just means that you will trust any certificate issued by router.asus.com as a valid certificate authority. I suppose some malicious site on the internet could use router.asus.com as their certification authority in hopes that people will have trusted it and will thus trust their cert. Seems pretty far fetched and unlikely though. You could avoid that by changing your router's domain to something else like "home.net" but that could cause issues if you ever tried to visit a site on the internet that has "home.net" as its domain. Maybe "home.zz" since .zz is not a valid extension but will work for local stuff. That's getting a bit paranoid though.
Basically any self signed certificate will have this same limitation - meaning someone did not pay for a certificate from one of the "big guys" and instead generated their own. Not something you would want to enable to some random internet site, but on a device you own/manage/trust it is fine.
Personally, I don't bother adding it to my trusted roots, if I visit it via HTTPS, I just click on the "proceed anyway". However stuff I have that uses my own domain that I own, I trust that root cert.
I believe you can get a free "letsencrypt" cert for access to the router, and they do use a trusted root authority for their certs. But probably not worth the effort unless you are accessing it from outside frequently (which isn't a good idea anyway without using a VPN).