I know this comma is correct, but why?

I read this on a news website:

'Mr Osbourne wants a golden age in British politics, something that is sorely missed'.

Would a semi-colon, a colon or a dash not be much less clunky?

Punctuation is a matter of style, so the comma isn't so much right or wrong as it is either in accordance with your manual of style or not. These manuals, which can differ in their recommendations, do not use "clunkiness" as a measure of punctuation's effectiveness. They strive to give rules for using the marks that will reliably enforce the proper parsing of text. I use the Chicago Manual of Style, which recommends that appositives like yours be set off by commas. An appositive is a renaming, here the renaming of a "golden age" as "something sorely missed."

A semicolon serves to separate independent clauses that are unjoined by a conjunction, and a colon serves to separate a list after the word following or a supporting or conclusory example. Neither of those is apt.

An em-dash can set off an aside, and if that's what the part after the comma is intended to do, i.e., to provide auxiliary commentary on the preceding part of the sentence, then such a dash would be appropriate.

• Mr Osbourne wants a golden age in British politics,
  something that is sorely missed.

comes from

• Mr Osbourne wants a golden age in British politics,
  which is something that is sorely missed.

by means of Whiz-Deletion, which deletes unnecessary Wh- is strings at the beginning of a relative clause (in this case, a non-restrictive relative clause), thereby -- in this case -- forming an appositive noun phrase, from the predicate noun phrase something that is sorely missed.

The comma is correct because non-restrictive relative clauses, and their reduced offspring,
must be separated in speech by comma intonation contours, and in writing by commas.