The earliest instances of "work from home" that a Google Books search turns up use the phrase to mean "work away from home." Thus, for example, in The Charters and General Laws of the Colony and Province of Massachusetts, we find this provision (enacted in September 1634) "relating to the impressments of labourers, horses, &c.":
Sect. 1. It is ordered by this court and the authority thereof; that in all publick works of this commonwealth, one magistrate and the overseer of the work, shall have power to send their warrants to the constables of the next towns, to send so many labourers and artificers as the warrant shall direct, which the constable and two other or more of the freemen, which he shall choose, shall forthwith execute, for which service, such magistrate and overseer aforesaid, shall have power to give such wages as they shall judge the work to deserve : provided that for any ordinary work, no man shall be compelled to work from home above one week together.
And from Reports of the Inspectors of Factories to Her Majesty's Principal Secretary of State for Home Department (1869):
When the mothers return to work, which they are obliged to do in a few days, where these [charitable institutions] exist not, or are not used, these nurslings are committed to any temporary care, and too frequently languish and die, or become diseased from improper diet. It is indeed true, that when mothers work from home, families must be neglected at home; and the voluntary extrusion of young mothers from factories and workshops, and even the limitation of married women in them, would be a national boon as well as a national gain.
The first uses of "work from home" in the sense of "work for a company without having a desk at its offices" that a Google Books search scares up take the form of classified ads in Popular Science magazine. These appear over the course of 24 years, between 1940 and 1963, and they appear more or less in the form of this one from the June 1940 issue:
BE Credit collection specialist. New method. Work from home. Full—part time. Unlimited opportunities. 70% profit. Details—franchise 25c. Modern Collections. 5000-D Cordelia Ave., Baltimore, Md.
Also interesting is this instance, from Central Office of Information [Great Britain], Reference Division, Home Affairs Survey, volume 2 (1951):
It is estimated that there are between 600 and 700 blind piano tuners in Great Britain, the majority of whom work from home.
Since piano tuning tends to be done at the customer's residence or business, it seems likely that "work from home" here means "work for themselves from out of their homes."
Still more to the point is this observation from International Labour Organisatioin, Documents of the Meeting of Experts on the Social Protection of Homeworkers (1991):
The United Kingdom survey clearly distinguishes between persons who work at home and persons who work from home, using it as a base for their work activities (e.g. salesmen, self-employed consultants and professionals of various kinds, freelance journalists and authors, etc.). The survey estimates a total of 658,250 for all home-based workers, with 229,790 (mainly women) working at home, and some 400,810 (mainly men) working from a home base.
Here "work from home" is an umbrella term that explicitly conveys the idea "work from a home base."
The first clear instance Google Books finds of "work from home" in the sense of "work at home for a company despite having a desk at its offices" comes from Datamation, volume 31 (1985):
Moving a dp shop from the city to the suburbs is one excellent way to cut costs and please employees. Another is the selective use of telecommuting, enabling some employees to work from home or elsewhere off-site for several days each week.
Because telecommuting is so closely tied to the earliest use of "work from home" in its modern sense, perhaps the phrasing appealed to people because it suggested the idea that the employee was "connecting to the office from home"—that is, the employee wasn't merely working "at home" but in some sense working "at the office from home."
In any event, the breakthrough year for "working from home" appears to have been 1994, the year Jeff Beamer published The Joy of Working from Home: Making a Life While Making a Living. The next year, another self-help book—Lionel Fisher, James Laroche & Sarah Fisher, On Your Own: A Guide to Working Happily, Productively and Successfully from Home—argued that "physically structuring a specific work zone within a residence should be a priority for homeworkers":
It means keeping where you live separate and distinct from where you work when home and office are under the same roof. Something that will help you do that is to remind yourself constantly: I work from home, not at home. There's a keen difference.
Regrettably, the authors never specify what that keen difference is.
I agree with the other answerers that "work from home" is now standard in many workplaces as a way of indicating days spent at home working on office business—whether telecommuting is involved or not. Still, though "work at home" may strike some hearers (as Tim Huynh suggests in his answer) as implying work on private projects at home rather that on company business at home, I don't believe that there is any firm basis for that distinction. I continue to use the phrase "work at home," and no one in management has yet sent me a note instructing me to "Stop working at home, and start working from home!"
Meanwhile, as far as I can tell, the vogue for "work from home" has not led to an upsurge in popularity for the phrase "work from the office."