I'm still getting my head around the terminology, but there are three kinds of things
1. Temperature "measurements" taken at a given site (modulo issues about multiple records, etc). I had thought that these were pretty much just measurements but some processing may have been done to them. These are the actual temperature and hence, in BE's model, contain both climate and weather effects. This is an "input".
2. A statistically-estimated/model-fitted/reconstructed absolute global temperature at that point at a given time along with similar values for climate and weather over all the observation stations, where they believe they've decoupled the original measurements into global temperature and effects of climate and weather. This is an output.
3. A gridded temperature anomaly, that is, values for what their model says the difference of the absolute temperature from "the global average" at regularly spaced locations where there may not have been an observation station. This is an output obtained by further interpolation of the output 2.
At various points they say they don't like 3, but because it's what all mainstream climate modellers produce they have to produce it to compare their results to others. They say that in terms of understanding what's going on in the world 2 is better. I had thought that all preprocessing before step 1 was doing was that, since they're not actually seeking out individual weather station reports but just big-ish collections that various people have already done, they' were just trying to throw out actual duplicates of reports from the same station. However, it seems like something additional has been done to the reports, which is a bit troubling when so unspecific.
I can't immediately find a sentence which says they are looking at historical recordings of temperature at weather stations. The closest I can find is this from the paragraph at the top of [this page](http://berkeleyearth.org/data) :
Source data consists of the raw temperature reports that form the foundation of our averaging system. Source observations are provided as originally reported and will contain many quality control and redundancy issues. Intermediate data is constructed from the source data by merging redundant records, identifying a variety of quality control problems, and creating monthly averages from daily reports when necessary.