How did words like rubbish, ribbon and cabbage get "BB"?

The spelling bb in rubbish, ribbon, cabbage is not entirely random because it is related to their pronunciations, as Decapitated Soul noted in the comments. The occurrence of a double consonant digraph such as bb is generally restricted in English to the position after a "short vowel" sound /æ, ɛ, ɪ, ɒ, ʌ/ or /ʊ/ (/ʊ/ is rare so I don't think any words actually exist with it before -bb-).

However, the double-consonant spelling pattern was not applied to all words with pronunciations of this type, as there are words like trouble, double, public, publish, robin whose standard present-day spellings do not have bb. Some words of French origin with short vowels were spelled with double consonant digraphs, while others were not; the reasons for this do appear to be fairly random. The Oxford English Dictionary notes old spellings with -bb- exist for double and robbin.

The pronunciations with a short vowel aren't completely straightforward to explain, but do seem to follow a general pattern. In English words from Anglo-Norman/Old French, it is fairly common for vowels in non-final syllables that were originally unstressed in French to be pronounced as short. Other examples are courage, savage, punish, lemon. However, there are also words where a long vowel developed in a similar context (bacon, label). For more detailed coverage of how the pronunciation of words like this developed, you can look at Otto Jespersen's Modern English Grammar on Historical Principles.