While modern English grammar tends to assume that 'the' is the definite article, we also use it as an odd and archaic adverb. In Old English þy was used as the instrumentive case of the neuter demonstrative -- or, to put it more clearly, the form of 'that' that indicates a causal relationship. As the language settled into the modern morass, þy melted into 'the', and no one much noticed.
This becomes much more intuitive given examples: the more the merrier, the sooner the better, the bigger they are, the harder they fall, the more I see, the less I like it. In all of these cases, the 'the ' is filling a role approximately equal to modern English's if/then combo; 'if more, then merrier", etc. It did not always change into modern form as paired the's, as in þy læs þe ('the less that'), a phrase that echos down the ages in phrases such as nonetheless and nevertheless. However, the adverbial 'the' does always pair with a comparative.
"Though the camomill, the more it is troden on, the faster it growes:
so youth the more it is wasted, the sooner it weares."
Henry IV, Part 1 (1598)
An interesting note to this; the Oxford English Dictionary makes the claim that these paired comparative clauses indicate a 'proportional dependence' -- that is, these the/the pairs can be roughly translated as 'to the degree that' / 'in that degree'. While that may be broadly true in spirit, it is most certainly not the sort of logical relationship most people mean when using these constructions; 'the more the merrier' rarely means "I hope a million people come!"; it just means that one more person is being welcomed into the group.