It appears to be a dedication to the author's family.
I haven't got any solid proof for this, but it's the solution I've seen put forward in a number of places, probably all ultimately deriving from J. H. McGivering, "Kipling and Son: a Successful Partnership", The Kipling Journal, December 1970, pp. 7-14. This article can be read either as a scanned PDF or in plain text at the Kipling Journal website. It's about the strong bonds within the Kipling family, especially between John Lockwood and his son Rudyard. From the very end of the article:
So much, then, for a quick look at two essentially lovable characters : the father dedicates his book
TO THE OTHER THREE
and the son never fails to show his family that gratitude and affection they so richly deserve; in fact that remark addressed to the Curator by the lama might well have been used by either Kipling to the other
"We be craftsmen together, thou and I".
This piece was also delivered as a talk at the Discussion Meeting of the Kipling Society on 15 July 1970, where apparently it inspired many questions some of which remained unanswered.
In a response to a question emailed to a Rudyard Kipling mailing list earlier this year, John Radcliffe said:
Surely the ‘other three’ members of the Kiplings’ family square, which consisted of Alice, Rudyard, Trix, and Lockwood.
This conclusion was further supported by Mary Hamer, who said:
Agreed. Bearing in mind also that Christmas number they put together, published as ‘Quartette’.
(Both Mary Hamer and John Radcliffe are officers in the Kipling Society.) Brian Harris, the querent in this email thread, also posted his findings on Quora.
The phrase "family square" was indeed used a lot for the four members of the Kipling family - you can find a lot of results online by searching the web for kipling "family square"
- so it seems reasonable that John Lockwood's dedication was, as many are, to his family.