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The importance of controlling the pH at 5.5 in the conversion of 6 to 25:

The pH is maintained at 5.5 to make sure the desired protonations and deprotonations can take place, leading to the final product. One key deprotonation that needs to occur, for example, is the deprotonation of the protonated amine in step 2 of the conversion of Intermediate 6 to Compound 25. This step is shown below.

The loss of this proton is necessary for the nucleophilic attack that occurs in the next step, which is crucial to the formation of the final product. At a pH of 5.5, this proton—which has a pka of 9.2—will be completely deprotonated (Keq>103) allowing for the product to be formed in good yield.

Soloway, S.; Lipschitz, H. J. Org. Chem. 1958, 23, 613.

Hall, H. K., Jr. J. Am. Chem. Soc. 1957, 79, 5441.

Bruice, T. C.; Schmir, G. L.; J. Am. Chem. Soc. 1958, 80, 148.

The importance of brine in the conversion of intermediate 6 to compound 25:

Brine (a saturated solution of NaCl) supplies the reaction mixture with counterions to stabilize any ionic intermediates formed over the course of the reaction.
In addition, by saturating the reaction mixture with chloride ions, brine—by Le Chatlier’s principle—also serves to prevent –Cl from leaving intermediate 6 as a leaving group (which could result in substitution and elimination side reactions).
In the absence of brine, an SN2 side reaction would be especially likely to occur due to the possibility of forming the following 6-membered ring.

Seiple, I. B; Su, S.; Young, I. S.; Lewis, C. A.; Yamaguchi, J.; Baran, P. S. Angew. Chem. Int. Ed. 2010, 49, 1095-1098.

Lane, T. J.; Quinlan, K. P. J. Am. Chem. Soc. 1960, 82, 3533.

Jones, R. A.; Katrizky, A. R. J. Chem. Soc. 1959, 1317.