When did Jews start keeping their beards? Did Moshe's generation have beards?
What is the historical development behind this phenomenon?
I assume that most people, besides Jews, also had beards, since it is a natural phenomenon that male humans have beards. At some point in time it became common for many men to shave them, and as mentioned by @Danny Schoemann, this was inconvenient for most Jews, and is still frowned upon by many for halachic reasons, so they are still more prevalent in Orthodox circles, even in communities that do not have any particular objection to shaving.
Asking when Jews started wearing beards is like asking when the Greeks started going uncircumcised.
The Torah commands us (Vayikra 19:27) not to use a razor on one's beard.
לֹא תַקִּפוּ פְּאַת רֹאשְׁכֶם וְלֹא תַשְׁחִית אֵת פְּאַת זְקָנֶךָ:
As a result, Jews can trim their beards but not shave them, as documented in Shulchan Aruch יורה דעה in סימן קפא - אסור גלוח הפאות
י: אֵינוֹ חַיָּב עַל הַשְׁחָתַת פְּאַת הַזָּקָן אֶלָּא בְּתַעַר, אֲבָל בְּמִסְפָּרַיִם מֻתָּר, אֲפִלּוּ כְּעֵין תַּעַר.
הגה: וּמִכָּל מָקוֹם נִזְהָרִים כְּשֶׁמִּסְתַפְּרִין בְּמִסְפָּרַיִם שֶׁיַּעֲשֶׂה הֶקֵּף הַגִּלּוּחַ בַּחֵלֶק הָעֶלְיוֹן מִן הַמְסַפְּרוֹת וְלֹא בַּתַּחְתּוֹן, פֶּן יַעֲשֶׂה הַכֹּל עִם חֵלֶק הַתַּחְתּוֹן וְהָוֵי כְּתַעַר (ת''ה סִימָן רצ''ה)
Only "recently" were electric shavers invented that possibly bypass the prohibition of using a razor on one's beard.
So in all likelihood, Jews started growing beards in Moshe's time, as soon as they were commanded to stop shaving with razors. As to whether they trimmed them, that does not seem to be documented.
According to kabbalistic sources, apparently, there's a spiritual advantage to not cutting one's facial hairs. According to them, the righteous ones - even in Moshe's time - would not have trimmed their beards. (ibid באר היטב ):
והאר''י ז''ל לא היה מגלח כלל לא בתער ולא במספרים לא בשום מקום כלל זולת בשיער שעל השפה המעכב האכילה היה חותך במספרים. גם היה נזהר שלא ליגע בזקנו שמא יעקר ח''ו ב' משערותיו ונמצא פוגם ועוקר צינור א' ח''ו) אבל הש''ך תפס דברי הרב עיקר והבו דלא לוסיף על החומרא עכ''ל:
To answer the title but not the body of your question, the Raddak who lived 1160–1235 in Narbonne, Provence, mentions that in his times in the lands where he lived the minhag was to not keep a beard. See Samuel 2 10 5. In fact his words there are apologetically trying to explain why in Dovid's times, when everyone had beards, it was embarrassing not to have a beard.
the Meam Loez in Bereishis says the lion, as king of the animals, was given a beard to give him a majestic look.
Likewise we can say for man who is the crown glory of creation and the one God chose for His service.
So according to this, it is proper for all men to have beards so it probably started from the first man (Adam) and was later contested by those who wished to deny the importance of man.