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In the quotation, the meaning of "how" is "that." What follows is the relevant definition from the OED (The Oxford English Dictionary) along with its examples:

III. In indirect questions and exclamations.

 10. With weakened meaning, introducing an indirect statement, after verbs of saying, perceiving, and the like: = That. Formerly frequently how that, and in modern dialect speech as how (see as conj. 26). see how still more or less calls attention to the manner.

c1000   Ælfric Joshua ii. 10   We gehirdon..hu ge ofslogon..Seon and Og.
13..   K. Alis. 1565   He..saide to the kyng, How his fadir hette Felip.
a1325  (c1250)    Gen. & Exod. (1968) l. 2732   We witen wel quat is bi-tid, Quuow gister-dai was slagen and hid.
c1405  (c1385)    Chaucer Knight's Tale (Hengwrt) (2003) l. 527   Hym thoughte how þt the wynged god Mercurye Biforn hym stood.
1548   Hall's Vnion: Henry VIII f. lvii   A letter was brought..certefiyng him how he was elected to be a Cardinal.
1571   in J. Cranstoun Satirical Poems Reformation (1891) I. xxix. 3   Seing quhow all erdly thingis wor subiect to mutatioun.
1611   Bible (King James) Ruth i. 6   Shee had heard..how that the Lord had visited his people.  
1709   I. Watts Hymns & Spiritual Songs (ed. 2) ii. 167   Sing how he left the Worlds of Light.
1750   T. Smollett Roderick Random (ed. 3) I. iii. 15   He was well informed as how, Rory was the best scholar of his age.
1801   Monthly Rev. 35 358   ‘If people knew as how they could talk, they would be obliged to work also.’
1843   Dickens Christmas Carol iii. 98   Bob Cratchit told them how he had a situation in his eye for Master Peter.

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