How did the term "esquire" come to be used for lawyers?

Esquire (in BrE) means not sir- it's a default title for somebody who doesn't otherwise have a title.

I suppose some American lawyer wanted to sound important and didn't understand the irony.

SUPPLEMENTARY TO coleopterist's ANSWER:

Instructions of Sir Henry St. George, Clarenceux, to Thomas May, Chester Herald, and Gregory King, Rouge Dragon Pursuivant, and Henry Dethicke for the 1682-83 Visitation of Worcestershire.

 In the allowance of titles, you shall enter the persons whose descents you take, with no other titles but such as they may justly and lawfully bear according to the Laws of Arms. And you shall allow the title of Esqr. to these and no other :—

 1. The heir male of a the younger sons of Noblemen.
 2. The heir male of a Knight.
 3. Officiary Esq^rs., viz., such who are so made by the King by putting on a collar of SS.,
    or such who are so virtute officii, without that ceremony, as the High Sheriff of a
    County, and a Justice of the Peace, during their being in office or commission ; with this
   caution, that you always enter the said office or qualification in speciall [sic] terms.

 As for Sergeants at Law, Doctors in Divinity, and dignified Prebends, you shall register them by those titles or qualifications only, but you shall accept them in quality as an Esq^r.
 Barristers at law you shall enter by that title, but you shall accept them as gentlemen only unless otherwise qualified to bear the title of Esq^r.

SOURCE: The Heraldry of Worcestershire (1873), by H. Sydney Grazebrook, Esq.; Of the Inner Temple, Barrister-at-Law, Author of the “Heraldry of Smith,” &c., &c. I, xlviii-xlix.

Grazebrook adds that “Barristers are now always accounted esquires, their right thereto being sufficiently attested by the fact that the Court of Common Pleas refused to hear an affidavit, because a barrister named therein was not called esquire ; and now all the Courts invariably require counsel to be so styled.--See Burke’s Patrician, v., 114, and Bythewood’s Conveyancing ii., 386, where the question of “Who is an Esquire?” is discussed at length.”

I'm an attorney, admitted to the bar over 20 years ago in California, and I practice in-house (not in a law firm). Depending upon the content and intended reader of my correspondence, I may deliberately sign my letters and emails with my name, followed by "Esq." because I want it to be obvious that I am a practicing attorney, and the correspondence may be protected by the attorney-client privilege. It's intended to be an obvious signal to the immediate reader - and in any potential litigation - that the correspondence might be protected as legal advice and should be reviewed before disclosure. My two cents. JT

American attorneys are Johnnies-Come-Lately who have unilaterally attempted to appropriate the term to identify themselves, but as has already been pointed out, this is a more recent event than the original meaning of esquire that continues through today. Untitled sons of nobles along with gentry below the rank of knight are identified by using Esq. after their names. Normally, this includes armigerous men without noble titles of their own. There are some men in the US who bear honorary British arms who appropriately use the Esq. suffix if they chose to do so and who are not attorneys. An since their rights to the suffix predates that of attorneys in the US perhaps they use it even more appropriately.

My speculation is that this usage in America may be related to Attorneys being considered officers of the court. The United States eliminated any recognition of nobility, peerage, or other titles granted by birth. When the United States became independent this was an important step as the United States sought to have citizens, rather than subjects. Therefore, my speculation is that this title survived in the United States, but only as it relates to an earned marker of a profession and not of parentage.

I note, as a supplement to StoneyB's supplement to coleopterist's answer, the following passage from Thomas Blount, The Academie of Eloquence Containing a Compleat English Rhetorique, Exemplified with Common-places and Formes Digested into an Easie and Methodical Way to Speak and Write Fluently According to the Mode of the Present Times (1654):

The youngest Sons of Viscounts and Barons are but Esquires, yet are honorable, and take place of all Baronets and Knights.

The eldest Son of a Baron is but an Esq during his Fathers life.

Esquire (comes from the French Escuier, in latin Armiger or Scutifer i. a bearer of Armes, or of a Sheild, and) is that Degree of Gentry, which is next to a Knight; It is conceived that at the first these Esquires were bearers of Arms to Lords and Knights, and thereby had their name and dignity: Now to be true Esquires according to the Law of Armes, they must either be Lords younger Sons, Baronets or Knights eldest Sons, members of Parlement, Sheriffs, Justices of the Peace, Serjeants at Law, Barristers at Law (yet the late Earle Marshall would not allow Barristers to be Esquires, but in the Act for Polemoney they were ranked by the then Parliament as Esquires and paid as Esq) or of some ancient family that has it by being heir to a Knight in the right line; Though now a dayes (I know not by what warrant) all Gentlemen that have but some considerable Estate in Lands, take that title upon them, when as the Estate (though never so great) adds no title. And that the title of Esq should descend from Father to Son (as the Estate of Gentry doth) is meer fabulous, saies Mr. Herne in his Glory of Generosity, p. 100.

Polemoney was evidently a poll tax exacted from prospective voters according to a sliding scale depending on each person's occupation and social status. Such a tax was enacted under Charles I in 1641 to pay for raising an army in England to quell uprisings in Ireland and Scotland, according to Wikipedia—but whether that is the Act for Polemoney that Blount mentions in the excerpt above, or whether he has in mind an earlier enactment of "polemoney" I cannot say.