Based on the book "Halachos of Niddah" by Rabbi Shimon D. Eider (first published in 1981), the origin of the modern Chassan and Kallah classes began shortly after the Holocaust. Historically, mothers would teach their daughters the relevant Halachos shortly before they were married. Unfortunately, after the Holocaust, there were many orphans who did not have anyone to learn the Halachos from, so formal 'classes' were established to fill this crucial function left in the wake of the destruction.
Additionally, there were a large number of people who came from irreligious families who were Chozer B'Teshuva, who could also not learn the Halachos from their family. The Teshuva phenomenon likely also added to the popularity of Chassan and Kallah classes.
Based on the above, it is likely that Chassan and Kallah classes became popular around the 1950's-1960's.
(Note that the question asks about classes that give "instruction and advice on relationship dynamics and intimacy". The main focus of the Chassan and Kallah classes are to learn the rules of Taharas Hamishpacha, per the linked question, but it's very likely that parts of the 'course material' evolved to include "instruction and advice on relationship dynamics and intimacy" due to the related subject matter. However, the main focus and intent of these classes is to teach the laws of Taharas Hamishpacha)
Quote from the book (from the preface, pages xvi-xvii):
During the period of the Rishonim and even until recent times,
daughters were taught by their mothers, the responsibility of Jewish
womanhood. Tradition was transferred - in the home - from generation
to generation. The halachos themselves, the severity of their
violation and the need to ask a Rav were inculcated in the hearts of
Jewish women by their mothers.[…]
The years of the holocaust and the upheaval it caused cast a gap in
the Jewish family. Mothers, who would have otherwise instructed their
daughters, perished at the hands of the latter day Amalek. As a
result, their daughters lost this vital link in the chain of
tradition. As a result, their daughters lost this vital link in the
chain of tradition.[…] In addition, we are blessed, nowadays, with the
new phenomenon of women […] who, through their own awakening, yearn to
observe these mitzvos impeccably. Therefore, in many Jewish
communities, this responsibility has been transferred or assumed by
Yeshivos for girls, kallah classes, and individual instruction by