How did the informant bird become a "little bird?"

Its modern usage might derive from Wagner's Siegfried as suggested by Wiktionary:

• In a Norse legend, Sigurd slew the dragon Fafnir and got a bit of dragon's blood on his tongue when he was roasting its heart. This immediately made it possible for Sigurd to understand what the birds were saying, and what they were saying was a warning that Regin would not keep his word, but instead planned to kill Sigurd.

• This was borrowed by Richard Wagner's Siegfried (Act 2), in which the main character comes to understand that the song of a small bird instructs him to steal a ring and helmet.

Common usage appears to be from the late 19th century according to Ngram about the time the Opera was released.

Ngrams finds a first hit dated to 1815. Searching for '"a little bird told me" "1815"' brings up Early American Proverbs and Proverbial Phrases, Bartlett Jere Whiting. It appears that Google has successfully picked up the date of the reference rather than the date of publication. That in turn leads to Modern Chivalry: Containing the Adventures of Captain John Farrago and Teague O'Regan, His servant, Volume 2, Hugh Henry Brackenridge (1815, link is to 1819 edition) which reads:

Hence the language of mothers to their children when they mean to say that they have got the information from a source they do not mean to explain, "a little bird told me of it."

Without spending a long time on context this pair of sources suggests that the phrase was reasonably well-known by then but not widely in print.

For a slightly different search "a little bird told" (without me) we find an earliest hit from 1764: the Gentleman's Monthly Intelligencer, Volume 33. The text there is Such is also the preſent phraſe—" A little bird told it me". Unfortunately very little context is available, but this again suggests an established (even if not common) use. Exactly the same sentence appears in The works, in verse and prose of William Shenstone, esq., published in 1773 after Shenstone's death (in 1763). The 1764 record seems to be a quote from Shenstone.

Shakespeare mentions bird 105 times in his works (search at www.opensourceshakespeare.org). None of these mentions bear any resemblance to the phrase in question. Of course just because he didn't use a phrase doesn't prove that it didn't exist in his time, however there are plenty of situations in his plays which it might be applicable. I suggest that the little bird variant really didn't exist much before the 18th century references.

Returning to ngrams for the presumed parent phrase "a bird told" we find nothing before the late 19th century. This suggests that previous uses were either specific (in the taxonomic sense) or used a different verb, though of course we don't have many digitised books from so long ago.

Searching for simply $BIRD told on ngrams gives the following earliest dates for various small birds, historically familiar in the UK and with long-established names:

wren    (nothing)
swallow 1857
sparrow 1789
robin   1840

These results may of course include some that are irrelevant

Accepting that the general figurative idea of a bird acting as a clandestine messenger is very old, how did the phrase arrive in its current form, a little bird told me? Why exactly did the bird become little?

Why do you assume that there was ever an expression involving a large clandestine bird messenger? I and the others attempting to answer haven't been able to think of such a prior expression, and I'd suggest there isn't one, at least not meaning inconspicuousness. That is, it seems to me that even if there were messenger storks, albatross, pelicans or eagles as a common expression, that expression probably didn't intend that they were inconspicuous. Or if it were, I'd think it might have the added meaning that it should be obvious (e.g. "An eight-foot bald eagle landed on my head and told me [something the writer thinks everyone should easily "see"]").

The idea of bird messengers is certainly very ancient and pre-dates English. Starting at the beginning, The Illiad features several examples of birds as messengers for the gods, and it seems interesting and relevant to your question that it was the interpretation of the type of bird, and perhaps what it did, that conveyed the message, rather than it speaking to the human receiving the message. Bird forms were also taken by gods to inconspicuously observe events (to avoid the notice of Zeus). Zeus himself tends to send eagles for more conspicuous communication.

Augury was a major component of Anglo-Saxon culture and still figures prominently in modern English culture through metaphor. One does not need to cite Norse or Greek mythology to find examples of cuckoos heralding the arrival of summer, or surprise entrances into the home foretelling good or bad luck (sparrow vs. swallow.)

The size of the bird is important, as others like Dronz have commented on. A little bird suggests a whisper, while a large bird is omenous. The stork foretold adultery; the raven foretold war and death; the eagle foretold victory.

I don't suggest this as a possible invention of the term, but the popularity of 'little bird' in the last century could be related to the relatively recent tradition of print media gossip columns on socialites, politics or movie stars, where 'a little bird told me' is an euphemism for 'a little spy told me' -- this phrase can be found extensively used both in print and online to reference unsubstantiated rumors about who is 'going with' whom, and also used in sports reporting to speculate about which footballer will transfer to which club in the off-season, etc -- 'a little bird' has become a staple of such articles. Influential gossip columnists might well have made it 'fashionable'.

Examples:

A little bird tells me that Actor A is in a relationship with Actress B.

A little birdie chirped in my ear that millionaire socialite and 'plastic surgeon to the stars' Dr.X will be running for Governor next year. If elected he has promised to give the State a facelift.

If you are willing to look extensively at online archives of gossip columns in print media you might well find a 'point of origin' (when 'little bird' was first used in a gossip context) and also interesting variations of the phrase in question.