What destroys the yoke?

The KJV of Isaiah 10:27 says:

And it shall come to pass in that day, that his burden shall be taken away from off thy shoulder, and his yoke from off thy neck, and the yoke shall be destroyed because of the anointing.

This week I was listening to a message where this passage was used as evidence that the anointing of the Holy Spirit (I guess roughly analogous to the Holy Spirit's power working in our lives) is what "destroys" the yoke of slavery. Problem is, I was reading the passage out of my ESV:

And in that day his burden will depart from your shoulder, and his yoke from your neck; and the yoke will be broken because of the fat.

A footnote on the word "fat" says "The meaning of the Hebrew is uncertain." The NIV and the NAS both go with "fat" as the proper interpretation. The RSV text, from which both the ESV and the NAS were adapted, seems to go with something totally different:

And in that day his burden will depart from your shoulder, and his yoke will be destroyed from your neck." He has gone up from Rimmon...

The Strong's definition of the word lists the meanings of the word as both "fat" and "oil" (as in anointing oil), with the KJV translating it most often as "oil." What I'm curious to find out is why, since "anointing" seems to be a translation that makes sense (at least to a layman) because of the association with olive oil, and "fat" makes no apparent sense at all... why would so many of the mainstream translations have gone with "fat"?