For example you would say "la angla" when referring to English. Why 1) is it an adjective and 2) do you use "la"? It doesn't seem to fit internally with the grammar. Is it adopted from another language or is there some deeper reason that I am missing?
It's merely an abbreviation: la angla = la angla lingvo. This way of naming languages has been totally conventionalised, so that you can find this structure even with languages where the language name is primary, for instance la latina = latino 'Latin'. Only with Esperanto you don't usually say "la esperanta".
It's short for la angla lingvo.
Without "la" it would be "an English language."
Just like in English, there are invisible (or implied) words.
- Mi parolas la anglan lignvon = Mi parolas la anglan.