I'm not an expert on Smith, but his LTV focusses on the labour that the consumer can save with their purchase. I know more about Marx, whose LTV (based on Ricardo) is based on the labour 'embodied' in a product by the producer. I'll try to explain this view in a different way than the other answer.The typical way that a price is determined for an elastical good looks something like:
Amortized fixed costs (buildings, machinery) - €0,50Variable costs (wages, resources) - €0,50Profit of x% - €0,50However, the contracter that built the building has made a similar calculation, so we can substitute 'building' with the entire list, which becomes
Amortized fixed costs ([machinery, wages, resources, Profit ], machinery) - €0,50Variable costs (wages, resources) - €0,50Profit of x% - €0,50We don't have to rewrite 'building' between the [] because we could do that infinitely and it would regress to zero. And we can do the same thing for machinery (which is made by companies with similar calculations) and for resources (which, if you repeat this process often enough, comes from the ground in lumber, ore, farms etc.). So the list becomes:
Amortized fixed costs ([[wages, Profit ], wages, [wages, Profit ], Profit ], [wages, Profit ]) - €0,50Variable costs (wages, [wages, Profit ]) - €0,50Profit of x% - €0,50As you can see, all that we are left with is labour and profit. Now, if we assume a healthy, functioning free market where competitors are ready to undercut any business with outrageous margins (even though we know from game theory that cartels can form spontaneously), it is clear that prices aren't "whatever subjective value some person attributes to the product" but a sum of all labour (its value) + the profit a capitalist wants to make by selling (its surplus value). If competition drives profits to near-zero, the price should be nearly equal to the value.
It is important to note that the authors you mention make this assumption of a functioning free market for goods meant for use/consumption. They never made the claim that the LTV holds for someting like land (capital), or gold or Supreme bricks (which have no use-value), or Van Gogh paintings (monopoly). For capital this is clear from the hard distinction that all the authors make between Profit and Rent; they have a seperate theory for land-rent, so the LTV does not directly apply to it. The requirement for functioning markets can be inferred from the following passage where Marx quotes Smith:
It suffices to say that if supply and demand equilibrate each other,the market prices of commodities will correspond with their naturalprices, that is to say, with their values as determined by therespective quantities of labor required for their production.
A full list of exceptions where the LTV does not apply (i.e. the price does deviate from the natural price) is found on https://en.wikipedia.org/wiki/Law_of_value.
