The common way is to decode the timing results into "marks" and "spaces", then interpret the mark/space sequences as bits, then check which button was pressed using its byte or word signature. Different protocols use different timings and encoding conventions, so identifying the protocol is useful to decoding the rest of the timing data.
This site has a good overview of IR remote control theory. Details of a few protocols:
SIRC,
XSAT. The XSAT timings look promising, but your timing data seems about 13% shorter (but this could just be the inaccuracy of the oscilllator of either your controller, or the IR remote).
If you cannot identify the protocol, you can still use a mark of ~480 us and a space of either 400 us for a "0" and 1200 us for a "1". There appears to be a quiet space of 1300 us before (I assume) a repeat signal starts, with a signal header consisting of a 6800 us mark and a 1600 ms space. These timings are based on the screenshot in you first post.