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They interpolate for latitude. For moonrise on Jan 23, N55 00, E 173° 37'  (- 11h 34m) we have the entries

N56 09 41 / 08

N54 09 31 / 09

I think in their examples they interpolate downward i.e. use start value N56 , end value N54.  Using  (X-Start)/(End-Start) * (data end - data start), where X is the latitude we want:   (55-56)/(54-56) * (31-41) = -1/-2 * -10 = -5   which we apply to the data start value:  41 - 5 = 36  minutes

They apparently also interpret the "diff" value:   1/2* (9-8) = 1/2  which we add to the diff start i.e. the 08:  08+1/2 = 8.5 rounded to 9

We have (note, we will keep track of the "diff" value of 9 by appending / 9 to our entries):

23day 09hr 36m / 9

we apply the correction, using longitude E173.6 ,  diff +9 .  They do not interpolate when using the F4 table, but use the closest values. In this case (180,10), and extract 10. Lon is east, so make it negative:  -10

23d  09hr 36m  / 9

               -10m /

23d  09hr 26m  / 9

next, adjust for longitude :  east lon means subtract 11h 34m:

23d  09hr 26m  / 9  =   22d 32hr 86m

                                     -     11hr  34m

                                    22d 21h  52m  / 9

We note that this is the *previous day* to the 23rd, and we want the time on the 23rd.  so add the time interval it will take the moon to rise the next day (hopefully this will be the 23rd, but it's possible the moon will next rise on the 24th).  The "diff" value in minutes is a correction to 1/2 day.  (I guess the tables work with half-days)  Anyways, for a full day, we add twice the diff value.  In other words, in this case, the 9 m means the moon will rise 1d and 18m later.  Add this to the date/time we found:

22d 21h  52m

  1d   0h  18m

22d 21h 70m =    23d 22h 10m

For moonset on the second example, the date/time found was the next day, so they subtracted  1day 40m to get the next earlier moonset.  But it turns out to be on the day before the 23, meaning the moon never sets on the 23rd when we are at the specified location.