Counting in hundreds is a survival of the usage in mediaeval England (or earlier), where a largely uneducated population dealt with quantitation in a few particular circumstances where great precision was not required. I suggest that its origin reflected the fact that it is simpler to express numerical ideas in terms of a single base quantity of suitable size (‘hundred’, ‘score’, ‘dozen’) in combination with small numbers — not more than two digits, and typically in the range 1–20 (for which distinct numbers exist in English). Counting in hundreds is also found in French and German, where other examples of numerical devices reflecting what I suggest is the same underlying psychology also occur.
The use of ‘hundreds’ in this manner dates from at least the middle ages. The Middle English Dictionary, for example, contains the following from ca. 1330:
“He miȝt him se.. ten hundred kniȝtes…”
“He [St. Francis] deide tweolf hundred ȝer and sixe and twenti riȝht Aftur ore louerdes burtime”
That, I hope, answers the “when?” and “where?” of the question. To deal with the “why?” (as far as the origin of the usage is concerned) it would seem necessary consider how quantitative concepts were expressed in a mediaeval society. The poster articulates the contemporary standpoint of one envisaging ‘1200’ in one’s mind, and then deciding whether to express that verbally as ‘one thousand two hundred’ or ‘twelve hundred’. However my thesis — which I can argue but not prove — is that for ordinary people in the middle ages (and earlier) there would be no choice: they would merely employ the natural method of counting in hundreds. This, I contend, reflects a general linguistic/psychological approach to quantitation by combining a single convenient base unit with relatively low and mainly integral numbers, which is easier for an uneducated population that generally does not require any greater precision than such numbers and their halves and quarters.¶
Examples of this general psychology of enumeration can be seen in the fact that single units were traditionally used for measurement. A person’s height was measured in feet, cloth was measured in yards, a horse’s height in hands, distances in furlongs or miles or leagues. Although an English mid-twentieth century school-child would have had to learn the relationship between such units from inches to miles, originally they were seldom used together; rather they were used individually for different purposes. Indeed more often than not there was no simple relationship between different units, and their rationalization to allow integral inter-conversion only came later in their existence. This has been discussed in relation to another SE question on the usage of yards for distance, and feet for height.
If we now turn to counting larger numbers, we can see how the orders of magnitude, ‘hundred’ and ‘thousand’ bear a formal relationship to units such as yards and miles.† (Note the use of ten hundred, rather than one thousand in the first citation above — the writer is thinking in a single register of hundreds.) This is supported by the fact that there were other non-base-ten orders of magnitude in use for lower ranges: dozen (12) and score (20). The dozen is still very much with us (and the word also exists in French and German), although the ‘score’ survives mainly in literature — the Bible, of course, and poetry into the nineteenth century (e.g. in Whittier’s Barbara Frietchie).
The poster may perhaps think that counting in hundreds is confined to English if he is a native speaker of a language such as Spanish or Italian, where apparently this is not found. However it is found in one romance language — French. Most obviously one can express dates in this manner — the year 1300 can be expressed as mille trois cents (one thousand three hundred) or treize cents (thirteen hundred). However counting in hundreds dates back to at least the fourteenth century in Les Chroniques de Sire Jean Froissart, a military history:
“Quand ils se trouvèrent tous ensemble, si furent plus de douze cents lances.”
(twelve hundred lances (men armed with lances or similar?))
More recently, Balzac used it in a financial context in Le Père Goriot (1910):
“Les choses sont comme cela chez vous, si l’on vous envoie douze cents francs par an.”
(…twelve hundred francs per year.)
and a recent internet search revealed that le douze cents is a contemporary idiom used by cyclists for mountains over 1200 metres.
My postulate regarding using various bases for counting is also seen in French: 20 is used as a base in the number 80 — quatre-vingt (four twenties) — presumably a remnant of ancient usage.
In German, too, counting in hundreds is longstanding. The following example is from Martin Luther in the sixteenth century:
“Ist das nicht greifliche Lugen g’nug, die mit zwölf hundert Jahren…”
(…twelve hundred years)
A later example is from a newspaper of 1805:
“…in einer Tiefe von zwölf hundert Klafter unter der Meeresfläche”
(to a depth of twelve hundred fathoms below the surface of the sea)
I am not sure how current this usage is today, but I was able to find one web page that had zwölf hundert Jahren in its title.
French and German have also employed the device of using a single convenient base, even after the adoption of the metric system. Thus, in contrast to the SI system used by scientists internationally in which only base units separated by three powers of ten are employed (microlitre — millilitre — litre), one finds cl used for volume (75 cl on bottles of wine, rather than 750 ml), hectares (100 ares — 10,000 square metres) for large areas, and hectograms (etti in Italian — 100 g) for weights of butter etc.
Why are literate and numerate English speakers (in both Britain and the US) still counting in hundreds in the twenty first century (another example, incidentally)? Traditional usage, perhaps — as in the examples given by @ChrisH — reinforced by our literary culture. However, also because it is often easier. Consider a bank clerk/teller counting out £1200 in £100 notes. He would likely start “One hundred, two hundred…” then shift to “three, four…” and end by saying “ten, eleven, twelve hundred”. Why change register half way through?
[¶ Of course, even in the middle ages, educated professionals such as architects, military engineers and accountants would work to greater precision. The Doomsday book (written in mediaeval Latin) has precise census numbers, but these are written in arabic numerals — they are not spelled out.
† The 1928 OED entry for hundred is interesting in this regard. It states that the word derives from the Old Norse, where there were two variants, one indicating 100 and the other 120. Although this latter seems to have been employed relatively rarely in English, it illustrates that ‘a hundred’ was seen as an entity in itself rather than a subdivision of ‘a thousand’. This is also supported by the fact that it takes the indefinite article like ‘dozen’ or ‘score’, whereas ‘ten’ does not.]