My summary: they claim (with evidence) that they found the sacred purple sarapis (tunic) of Alexander the Great, and possibly some of his other things.
I'll go a bit further and say that they don't claim it's Alexander's tomb, but someone that was buried with Alexander's artifacts (namely, his brother)
This is correct. Alexander is burried in Alexandria, Egypt. This discovery means that some of his artifacts were inherited by one of his siblings, moved back to Greece and burried with them.
AFAIK, the current location of Alexander's tomb is not known.
The exact location is not known but there is strong consensus amongst historians that he's burried in Alexandria.