The linked article also claims that the NOAA "deleted" the post-1960 data from a data set (briffa2001jgr3.txt) put together for an article that doesn't use these data. In other words, what really happened, is that Briffa and his collaborators created a data set only containing the data they used, and this derived data set was archived by NOAA. That's hardly nefarious or even the least bit worrisome.
Now to the heart of the issue: Why was the post-1960 not used after 1998/2000? Why do all of the e-mails talk about "hiding the decline" and the code talk about stopping in 1960 to "avoid the decline"? The answer, it turns out, is right in the very Briffa 1998 paper your link mentions (but fails to cite). It's title is Reduced sensitivity of recent tree growth to temperature at high northern latitudes and it's about how temperatures reconstructed using Briffa's MXD (maximum latewood density) data diverge from actual temperature readings after 1960. The implication of the word "reconstructed" is very important to understand: Briffa's MXD data don't show an actual decline in temperature. They show a decline in a particular measurement of tree-ring density that had, until 1960, been a good indicator of actual temperature. All that's going on here is that data that is known to be inaccurate is not being used.
It's no big deal.