A quick recap on its movement so far:
The Anomaly was discovered about a week ago near Oochorrs UF-J c11-0 (on the far side of Barnard's Loop, 2000ly from Sol). It appeared to be approaching from the direction of (permit-locked) NGC 2264 Sector RE-Y c14-0, 73.3 lightyears away, estimated to be moving around 0.7ly/h. We don't know where it was before then, or when it first appeared.
It finally reached UF-J on Sept 1 18:17 UTC, and began heading to a new system, Oochorrs CS-F c13-0, 57.19 lightyears away.
It reached CS-F on schedule at Sept 5 05:09 UTC, and began heading to Oochorrs BS-F c13-0, 10.50 lightyears away.
Around 12:30 UTC the same day, it suddenly massively increased its speed to an estimated 17-18ly/h, reached BS-F at 12:55 UTC, and began heading to HD 38291, 68.71 lightyears away.
After getting almost half-way there within hours, it slowed down again (we think it's now moving roughly 1ly/h), and should reach HD 38291 on Sept 7 02:06 UTC if it doesn't change speed again.
Today, Galnet finally posted an update referring to the phenomenon as an unexplained "flare":
It may be of interest that the anomaly will closely pass by another system today, one that it is not entering directly. Oochorrs FY-D c14-0 lies almost on the projected path of the Anomaly, being only 0.7 lightyears away from the direct line between BS-F and HD 38291. The Anomaly is expected reach this closest approach around 20:42 UTC today. I'm hoping the view might even be better than the head-on view from HD 38291, since the Anomaly should rapidly move across the sky.
This screenshot shows the path on the galaxy map:
Source: https://www.reddit.com/r/EliteDangerous/comments/x75x3x/summary_on_the_anomalystargoidetc/