As another user has already brought up, 'transgender' is mostly prefered over 'transsexual' within the community, the latter of which many consider to be offensive. We realize that proper terminology can be difficult to keep up with and don't believe any ill-will was intended though, so ask that users keep that in mind but remain focused on the question at hand. If you have further questions about terminologies within the LGBT+ community, we'd ask that you take those to the appropriate subreddit. If you have questions about moderation of the sub, modmail is always open.
A note on the process of historical research here: In cases like this, it is difficult to give a complete picture because we lack sources that describe the perspective of transgender individuals beyond a few famous cases. I can and will describe the legal situation but in terms of how a transgender individual save a few actually perceived how they were treated, it is difficult to say.
What can be said is that since 1910 owing to the efforts of Magnus Hirschfeld, a physician by trade and founder of sexology in Germany, the legal situation in Germany was as such that transgender individuals could obtain a so-called "Transvestitenschein" (a transvestite certificate or pass) indicating towards authorities of the state that they were transgender individuals and therefore avoiding for example arrest for public nuisance (e.g. a man wearing women's clothing would have been that to German law enforcement in the 1910s).
Hirschfeld in general advocated for gay, lesbian, bisexual, and transgender rights, specifically to decriminalize male homosexuality in Germany. In that regard he was not successful since §175 remained in effect in the GDR until 1988 and in the unified Germany until 1994 (albeit with significant changes). When he founded his Institute for Sexualwissenschafte in 1919 he continued to advocate – one could say in a sense he pioneered – a different understanding of transgender.
At first, he subsumed what he described as the instinct or urge to disguise one's gender (Hirschfeld still uses sex because while his writings certainly acknowledge the social aspect of sex, he didn't have the concept of gender at his disposal) as Transvesiteism, which in today's understanding covered both cross-dressing as well as transgender individuals. In 1925 however, he reconfigured his theory and coined the term "transsexual" in essence describing that an individual can feel like it is born in the wrong body and wants to change their gender/sex. This was something Hirschfeld didn't perceive as a pathology but in line with his theory that there are about 64 possible types of sexual intermediary ranging from masculine heterosexual person to feminine homosexual person.
In the meanwhile, medical procedures, especially in hormonal therapy were invented and introduced that allowed for people to indeed change their sex. This also became legally possible in the Weimar Republic though in order to official change one's status from male to female or female to male in your official documents, you had to appear before a judge, undergo evaluation and an operative sex change. One of the first and probably most famous cases was that of Lily Elbe. Born Einar Wegener in Denmark, she went to Germany in 1920 to undergo the still experimental procedure of a sex change operation and lived the rest of her life as Lily.
The so-called Tranvestitenscheine were easier to obtain but those didn't change your status in the eyes of the state, they only indicated to law enforcement that you were a F2M or M2F or cross-dressing individual but even for those, you had to register with your local police department.
While this might seem fairly progressive from our viewpoint, as I said, what is missing is the perspective of the people who were affected. These procedures, especially appearing in front of a judge and undergoing medical evaluation could be a traumatizing experience and could represent a very high barrier for people simply to realize who they actually were. Similarly, for those who did want to live as a man or woman but were born a woman or man, yet didn't want to undergo a medical procedure in order to realize what their true identity was, there was no possibility to do so.
Also, while Hirschfeld for his studies presented at least 70 cases of transgender individuals, from official records, we know of very little actual examples of people undergoing the procedures or have written down accounts of their lives.
While Hirschfeld was scorned by the Nazis and had to flee Germany fearing persecution as a Jew and a sexologist, the sitaution of transgender individuals in Nazi Germany still remains underresearched. As I wrote in my answer here, historical research so far has turned up about 25 biogrpahies of transgender persons in the Third Reich who have official documentation attached to their names and of those seven persecuted while the others did have problems with the authorities but were for example not send to a camp.
Sources:
Peter Gay: Weimar Culture: The Outsider as Insider. 1981.
Peter Jelavich: Berlin Alexanderplatz: Radio, Film, and the Death of Weimar Culture 2009.
Volker Weiss (2010), „Eine weibliche Seele im männlichen Körper; Archäologie einer Metapher als Kritik der medizinischen Konstruktion der Transsexualität“. Dissertation FU Berlin.
Rainer Herrn (2013), „Transvestitismus in der NS-Zeit – Ein Forschungsdesiderat“. Z SexFo 26.
Ilse Reiter-Zatloukal (2014); "Geschlechtswechsel unter der NS-Herrschaft. 'Transvesttitismus', Namensänderung und Personenstandskorrektur in der 'Ostmark' am Beispiel der Fälle Mathilda/Mathias Robert S. und Emma/Emil Rudolf K."; Beiträge zur Rechtsgeschichte Österreichs, Bd 1-2014.
Elena Mancini: Magnus Hirschfeld and the Quest for Sexual Freedom: A History of the First International Sexual Freedom Movement 2010.