They advance in the same manner as the Vestiges of Divergence.
The description of “Betrayer Artifact Properties” (Explorer’s Guide to Wildemount, p. 274) states:
The Arms of the Betrayers advance in power in the same manner as the Vestiges of Divergence.
The rules for advancement of a Vestige of Divergence are spelled out explicitly on page 271 of EGtW. Of particular note is the property that their advancement is entirely up to the DM. The Arms are sentient. You can’t decide to advance them; they decide for themselves who they grant their power to.