From the 1969 ARRL "The Radio Amateur's Operating Manual"

The traditional expression "73" goes right back to the beginning of the
landline telegraph days. It is found in some of the earliest editions of
the numerical codes, each with a different definition, but each with the
same idea in mind - it indicated that the end, or signature, was coming
up. But there are no data to prove that any of these were used.

The first authentic use of 73 is in the publication 'The National
Telegraph Review and Operators' Guide,' first published in April 1857.
At that time, 73 meant "My love to you!" Succeeding issues of this
publication continued to use this definition of the term. Curiously
enough, some of the other numerals then used have the same definition
now that they had then, but within a short time, the use of 73 began to
change.

In the National Telegraph Convention, the numeral was changed from
the Valentine-type sentiment to a vague sign of fraternalism. Here, 73
was a greeting, a friendly "word" between operators and it was so used
on all wires.

In 1859, the Western Union Company set up the standard "92 Code." A
list of numerals from one to 92 was compiled to indicate a series of
prepared phrases for use by the operators on the wires. Here, in the 92
Code, 73 changes from a fraternal sign to a very flowery "accept my
compliments," which was in keeping with the florid language of that era.

Over the years from 1859 to 1900, the many manuals of telegraphy show
variations of this meaning. Dodge's 'The Telegraph Instructor' shows it
merely as "compliments". The 'Twentieth Century Manual of Railway and
Commercial Telegraphy' defines it two ways, one listing as "my
compliments to you"; but in the glossary of abbreviations it is merely
"compliments". Theodore A. Edison's 'Telegraphy Self-Taught' shows a
return to "accept my compliments". By 1908, however, a later edition of
the Dodge Manual gives us today's definition of "best regards" with a
backward look at the older meaning in another part of the work where it
also lists it as "compliments".
"Best regards" has remained ever since as the "put-it-down-in-black-
and-white" meaning of 73 but it has acquired overtones of much warmer
meaning. Today, amateurs use it more in the manner that James Reid had
intended that it be used - a "friendly word between operators".