The Labor Theory of Value is not itself a scientific theory. It is, instead, a "philosophical" or "historical" theory about more-or-less scientific theories of political economy. The LTV does not by itself predict the world; it is, instead, a way of making our economic predictions intelligible and meaningful. It asks the question, what do we mean when we're talking about economics? What is a "price"? What is "money"?
The LTV is not an "unscientific" theory: it does not (like many theologies) make covert scientific predictions that the real world does not bear out. It is, instead, a "metaphysical" theory, but only in Popper's sense; there's nothing transcendental or mystical about the LTV. It's about that most ordinary and material activity: real material human beings doing real material work producing real material goods and services. Indeed, the LTV denies transcendent or immaterial elements in economics.
The LTV starts from the premise that the only thing human beings have that we can trade is our labor. Our labor is (collectively and individually) the only thing that's ours. We must, of course, work upon nature; without a natural world, our labor would have nothing to work on, but nature itself is not ours; only our own labor is truly ours. Even Locke agrees: what fixes a part of nature as someone's property is precisely her labor operating on that part of nature, and she may expropriate a part of nature only so long as there is enough of nature sufficient so that others can expropriate their own part of nature. If nature is freely available, one cannot trade it, just as no one can trade air; all we have left to trade is our own labor.
The only free trade is a trade of equivalent exchange-values: one unit of labor embodied in a commodity for one unit of labor embodied in another commodity. The entire reason that trade actually does occur is because of the division of labor, the use-values of the two commodities are asymmetric: first, my commodity is of more use to you than to me, and your commodity is of more use to me than to you; second, my commodity is of more use to you than your own commodity, and your commodity is of more use to me than my own commodity. The asymmetry of use-values makes trade possible; the equivalence of exchange-values is what makes trade fair.
Marx did not invent this idea. Adam Smith and David Ricardo invented the LTV, and in Capital, Marx credits Benjamin Franklin with the idea that all we can do is trade our own labor. Marx's innovation is the distinction between labor power and actual labor: the ability to work for a period (i.e. a day), and the amount of work that can be actually done in a day. Labor power is (under capitalism) a commodity: it has an exchange-value, the Socially Necessary Abstract Labor Time (SNLT) required to create a working day, and a use-value, the total amount of Abstract Labor Time (LT) actually created. Marx labels the difference between the cost (in SNLT) of labor power and the amount of labor expended as "profit".
There are several objections to the LTV, most of which Marx addresses in Capital.
The first objection is that that if two people (or firms) create a commodity, and the first person expends half as much labor (in total, including labor used to produce raw materials, intermediate goods, amortized capital and administration) than the second, then the second person's commodity is absurdly worth twice as much as the first person's. To overcome this objection, Marx introduces the modifier, "socially necessary." If for example, in the material social and technological context of a society, the demand for coats can be satisfied by people working at most two hours per coat, then the exchange-value of a coat is two hours. If your firm takes four hours to make a coat, well, too bad for you. There are enough coats being produced using two hours of labor per coat that no one is forced by the scarcity of coats to pay you four hours. If, on the other hand, your firm can create a coat in only one hour, well, good for you. Because there are not enough coats that can be produced using only one hour of labor, people are forced to pay two hours of labor for a coat. Your firm will make an "economic profit" of an hour of labor per coat. (The capitalist who can capture such an economic profit is thus exploiting not only his workers, but his customers as well.) Economic profit is different from profit; even a producer who requires two hours of labor per coat, and exchanges them for two hours, will make a profit, because she will pay her own workers only for the cost of their labor power, which will be less than the total amount of labor they actually expend making coats.
The second objection is that actual labor (like use-value, as Marx argues elsewhere in Capital) is radically heterogeneous. The actual tasks performed by one individual laborer making one commodity are qualitatively different from those performed by another laborer making another commodity. Marx admits this, but what is still equivalent is the time a laborer expends, abstracted from the specific tasks he performs. (Of course, the worker still has to actually perform those tasks, and perform them efficiently, but the specific nature of the tasks does not enter into exchange value).
Related to the second objection that even labor abstracted from the heterogeneity of specific tasks is still heterogeneous. People differ in relative skill, talent, the intensity of their labor, etc., again, even abstracted from the specific tasks. Marx himself mostly handwaves this objection away, talking about averages and aggregates. However, this objection does not seem that important. First, Adam Smith argues that there is actually little difference in inherent ability; the difference between a philosopher and a laborer arises because of education and social situation, not inherent ability. Second, what matters is the socially necessary (marginal) labor time. If the "last" person hired to make a commodity were an inefficient slacker who takes twice as much time to produce the commodity as his most efficient co-worker, well, then that sets the socially necessary labor time. If the firm could hire someone more efficient, they would do so; if they could not sell the commodity for the price implied by the slacker's time, they would not hire him. Even if there really are innate differences in ability, we're still interested only in ability at the margin, which is measurable.
A third objection is that firms do not actually account for labor and labor power in the way corresponding to the LTV. Marx argues that this discrepancy is by design. If our accounting were honest, then no one would stand for capitalist exploitation; instead, the exploitation has to be carefully hidden for a capitalist society to work at all. Marx argues that this obfuscation is discernable: he argues that we (absurdly) think we have a social relationship with objects, when in reality producers of different commodities have a social relationship with each other.
There are other, deeper objections to the Labor Theory of Value, which I will (might) discuss in a further post,