People have suggested that the Anchor is holding the bearer in place, and allows the bearer to slip between the Fade and the waking world on a whim. Something about that doesn't seem quite right, though.
The Anchor came from that glowy green orb that belonged to Solas (or perhaps was originally Mythal's). Given what we know of Solas and his goals, he has a lot of interest in the Fade and the Veil, and of course we hypothesize that he could be trying to take down the Veil. But for now, he is strengthening it, so that spirits don't get warped into demons from the pain and shock.
What does this have to do with anything?
I think the Anchor may be holding the Veil in place. It was originally part of the orb; Corypheus wanted it separated from the orb so that he would be able to use this tremendous power for himself, and get to the Black City. Some of Solas's hand gestures suggest to me that originally, the Anchor was attached to Fen'harel (so why did he put it in an orb? Safekeeping?). All of those elven artifacts strengthen the Veil, but without the power of the Anchor, it comes loose and unravels.
This would also suggest that the orb's original location--or the location of another orb--would be within Skyhold (due to the past suggestion that there are two, because an ellipse requires two foci). So, either Solas woke in Skyhold, or sooner or later he's going to have to return for the orb that got left behind.
Either way, the shiny green orb that is now broken contained the Anchor, which is now attached to the Inquisitor's hand. The Inquisitor did not BECOME the Anchor. She holds it, and with it, Solas says, "the key to our salvation." Whose salvation? Probably, the ancient Elvhen, or elves in general. Though Solas may truly believe that everyone will benefit from... whatever it was Solas wanted to do with the Anchor.
If I'm right, then having the entirety of the Anchor's power means that the Inquisitor has the ability to either continue to uphold or completely remove the Veil, though he/she will probably need to go through some extra step or steps that we don't know yet to unlock that capacity--a failsafe, if you will. Solas needed the Anchor for whatever he's planning to do--freeing the Old Gods and/or Creators, or removing the Veil, or both, or neither.
We also know that, had Solas gotten the orb, we would have seen whatever the truth was. Perhaps he would have made another Anchor, or perhaps he would have been able to remove the Anchor from the Inquisitor.