You shouldn't look at the phenomenon as originally "a double surname", but rather as having several surnames.
There are many ways and reasons for one to acquire a second or third surname. England was in many ways not that much different from the Continent, I believe. In Mediaeval times, one could, for example, be lord of both Aragon and Castile, and use the name "Ferdinand of Aragon and Castile" in official correspondence (although the higher the rank, the more names one usually had and has). Prepositions are still used, like the famous German "von und zu", theoretically indicating that one came from a certain area and/or rules over it, and still resides there; "von und zu Franckenstein" is short for "von Franckenstein und zu Frankenstein". Or one could be known as "of Bavaria", but of a cadet branch residing at Heidelberg, and call oneself "Henry of Bavaria Heidelberg". This could later be shortened to "Henry Bavaria Heidelberg". The use of a specific spelling or composition of one's name was not important in the Middle Ages, not like now.
An important reason to add a second "barrel" to the name of a commoner is to prevent the name of one's mother's family from dying out. In Dutch law, this is still the only situation in which one is legally allowed to add a second name to one's name. I suspect that this is also behind many of the well known English double-barelled names that are old but not noble (or were only recently ennobled). (My mother's (non-noble) family got their second barrel when the name of some great-great-grandmother was about to go extinct. This was under the Napoleonic occupation, whose laws forbade adding names; so the son was officially entered into the register as having his mother's name as one of his first names, later to be insidiously turned into a surname.)
In modern times, these double or triple or quadruple names have been formalised. Conventions have been established about e.g. hyphenation, varying across Europe. In Holland, double-barrelled names are never hyphenated; a hyphenated name is always that of a married woman and not inheritable (I'm sure there exceptions, though, as there always are). In Germany, double-barrelled names are normally hyphenated, I believe. Practice in England seems to vary. A double-barrelled surname is now legally treated as together being a single name in most of Europe (Iberia is of course different), regardless of hyphens or spaces.
One "barrel" is normally considered the "core" of the name: most people I know use only the core barrel in informal situations, such as when introducing oneself at a party.