The here is an adverb. According to the definition at Merriam-Webster,
(2.1) THE - adverb.
"than before : than otherwise". Used before a comparative:
none the wiser for attending
(2.2)
a : "to what extent"
the sooner the better
b : "to that extent"
the sooner the better
According to the definition at Wiktionary:
(2.2) THE: adverb.
With a comparative, and often with for it, indicates a result more like said comparative. This can be negated with none.
It was a difficult time, but I’m the wiser for it.
It was a difficult time, and I’m none the wiser for it.
I'm much the wiser for having had a difficult time like that.
The modifies the adverb more and they together form an adverbial modifier that modifies the verb doubt.
According to Wiktionary, the etymology is as follows:
From Middle English, from Old English þȳ (“by that, after that, whereby”), originally the instrumental case of the demonstratives sē (masculine) and þæt (neuter).
Imagine that þȳ has survived in the Modern English:
John: It was me who extinguished this fire. The rain has nothing to do with this.
Paul: I doubt this.
John: Yes! I extinguished it þȳ blacke hose! (with that black hose; points at the hose)
Paul: I doubt this þȳ more! (I doubt this [to that extent] more; see Merriam-Webster's def. 2.2; the extent of his doubt has increased "by that amount" due to John's lie - by some unmeasurable amount that corresponds with Paul's astonishment at John's lie)
John: Why?
Paul: The hose is not connected to water supply!
Here, þȳ is used to enhance the meaning and is roughly synonymous with even.
This use of the as adverb occurs in comparative constructions, where the modifies some comparative word like "more, less" etc. In some such constructions, we can drop the:
Her mood wasn't any (the) less bright for her loneliness.