There is, in fact, no law or codified legal definition passed by parliament of human dignity (Menschenwürde) in Germany. This is by design. Compare with the text of article 4, section 3:
Niemand darf gegen sein Gewissen zum Kriegsdienst mit der Waffe gezwungen werden. Das Nähere regelt ein Bundesgesetz.
No person shall be compelled against his conscience to render military service involving the use of arms. Details shall be regulated by a federal law.
Translation as given by Wikisource
In this article, it is explicitly mentioned that details regarding the right not to serve in the military against one's conscience are to be regulated by a federal law. Article 1 (like most of the basic rights articles 1 through 19) lacks that explicit statement.
Of course, the lack of an explicit statement requiring regulation by law does not mean something cannot be regulated by law. Nonetheless, throughout the history of the Federal Republic of Germany no government felt the need to define the term by passing an appropriate law.
Instead, human dignity is to be understood as a general guideline that the government and parliament have to interpret in order to legislate accordingly without violating the spirit of the article. The closest there is to a legal definition of human dignity is the content of the following articles 2 through 19, where they codify human and citizen rights (such as freedom of speech, right to private ownership, gender equality, etc.).
On the other hand and maybe somewhat surprisingly, it is indeed the judiciary that is most active in defining the scope and limitations of human dignity; more precisely it is the Federal Constitutional Court. This has a number of responsibilities but the two that stand out in this context are the Constitutional Complaint (Verfassungsbeschwerde) and Regulation Control (Normenkontrolle). The second is the request by a court of law, the Bundestag or a state or the federal government to confirm that a law complies with constitutional requirements while the first -- which makes up upwards of 90 % of cases the court has to deal with -- is a complaint by a person to investigate whether any branch of government has violated their constitutionally protected rights.
In important cases the court has to decide, it will often elaborate how and under which circumstances specific acts or laws violate the spirit of Article 1, human dignity. One example was its ruling on the Luftsicherheitsgesetz (Air Security Law) of 2005 (link in German). Briefly, the law which was written based on the experience of the 2001 terror attacks in the US included a provision to shoot down an aeroplane which was being used as a weapon against civilians as a last resort. Among other reasons, it ruled that this law violated article 1. It elaborated that a core element of human dignity is being treated as a subject, permitting one's own choices of action. In the case of a plane being shot down, the passengers and crew would, however, be degraded to objects at the hand of the state as they would have no chance to influence their fate. Copied into the quotation block below is the relevant part of the original ruling in German which would be too complicated for me to translate.
Was [die Verpflichtung zur Achtung und zum Schutz der Menschenwürde] für das staatliche Handeln konkret bedeutet, lässt sich nicht ein für allemal abschließend bestimmen (vgl. BVerfGE 45, 187 <229>; 96, 375 <399 f.>). Art. 1 Abs. 1 GG schützt den einzelnen Menschen nicht nur vor Erniedrigung, Brandmarkung, Verfolgung, Ächtung und ähnlichen Handlungen durch Dritte oder durch den Staat selbst (vgl. BVerfGE 1, 97 <104>; 107, 275 <284>; 109, 279 <312>). Ausgehend von der Vorstellung des Grundgesetzgebers, dass es zum Wesen des Menschen gehört, in Freiheit sich selbst zu bestimmen und sich frei zu entfalten, und dass der Einzelne verlangen kann, in der Gemeinschaft grundsätzlich als gleichberechtigtes Glied mit Eigenwert anerkannt zu werden (vgl. BVerfGE 45, 187 <227 f.>), schließt es die Verpflichtung zur Achtung und zum Schutz der Menschenwürde vielmehr generell aus, den Menschen zum bloßen Objekt des Staates zu machen (vgl. BVerfGE 27, 1 <6>); 45, 187 <228>; 96, 375 <399>). Schlechthin verboten ist damit jede Behandlung des Menschen durch die öffentliche Gewalt, die dessen Subjektqualität, seinen Status als Rechtssubjekt, grundsätzlich in Frage stellt (vgl. BVerfGE 30, 1 <26>; 87, 209 <228>; 96, 375 <399>), indem sie die Achtung des Wertes vermissen lässt, der jedem Menschen um seiner selbst willen, kraft seines Personseins, zukommt (vgl. BVerfGE 30, 1 <26>; 109, 279 <312 f.>). Wann eine solche Behandlung vorliegt, ist im Einzelfall mit Blick auf die spezifische Situation zu konkretisieren, in der es zum Konfliktfall kommen kann (vgl. BVerfGE 30, 1 <25>; 109, 279 <311>).
quoted from Wikipedia
Without having to understand the entirety of what has been written in that quote, note that there are cross-references to other rulings the court has made on how to interpret human dignity; they begin with vgl. BVerfGE.
The ability of the Federal Constitutional Court to elaborate what exactly constitutes human dignity even though Germany is a civil law country is regulated by law, specifically the Bundesverfassungsgerichtsgesetz (BVerfGG) or Act on the Federal Constitutional Court which regulates the details of how the court operates in accordance with Article 94, Basic Law. § 31 BVerfGG regulates that decisions of the court are binding to executive, legislative and judiciary powers and that in certain enumerated cases these decisions have the force of law. Essentially, this makes the highest court of a civil law country a type of common law precedent-setting court.
As minor asides, it is worth noting that precedent also matters in civil law countries (i.e. it is generally expected for later cases to be decided in the same way as previous ones unless the situation is markedly different) and that the Federal Constitutional Court is not part of the appeals process. One cannot appeal one's case all the way to the Federal Constitutional Court. If however, one successfully claims that another court violated one's constitutional rights in a Constitutional Complaint, the FCC may annul the ruling in question including the entire chain of appeals and resubmit the case to the lowest court anew.