Please explain/summarize "Lawrence v. Texas." Why is jurisprudence important in this case?

The Supreme Court's decision in Lawrence vs. Texas was important because it invalidated previous Texas law about sodomy.
Jurisprudence is the science of law. Basically, jurisprudence determines the legal basis for a set of laws. Essentially, it determines the rules that should form the basis for these laws and also sets a precedent for how similar future cases should be decided.
Here is a short summary of Lawrence vs. Texas: On September 17th, 1998, John Geddes Lawrence invited Tyron Garner and Robert Eubanks to his home for a get-together. Garner and Eubanks were once lovers, and this fact will prove significant in what happens next. During the course of the evening, Eubanks became extremely upset after suspecting that Lawrence was flirting with Garner (his former lover). For his part, Eubanks had never gotten over his feelings for Garner.
Eubanks left to purchase something at the store. However, he soon decided to call the police and to report that someone was brandishing a gun at Lawrence's apartment. Eubanks knew that he was making a false report, but he did not care.
When police got to Lawrence's home, they found Lawrence and Garner in a compromising position. Both were promptly arrested and, later, tried in court for falling afoul of Texas's sodomy laws. Under the Texas anti-sodomy law, it was considered a Class C misdemeanor to engage in homosexual intercourse.
The two decided to sue for their rights and were represented by Lambda Legal in court. Lawyers from Lambda Legal argued that the Texas laws were unconstitutional and that they violated the equal protection clause enshrined in the 14th Amendment. The case was fought all the way to the Supreme Court.
Lambda attorneys asked the Supreme Court to determine whether the legal basis for the Texas sodomy law was unconstitutional. They basically asked the court to address 3 questions:
1) Whether the Texas sodomy laws violated the equal protection clause in the 14th Amendment.
2) Whether the petitioners' criminal convictions regarding consensual same-sex relations violated their privacy and liberty.
3) Whether Bowers vs. Hardwick should be overruled.
In the end, the Supreme Court decided for Lawrence and Garner. In a 6-3 decision, the court decided that the Texas sodomy law was unconstitutional. The court also overruled Bowers vs. Hardwick. Judge Anthony Kennedy wrote the majority opinion. He argued that liberty extended to both heterosexual and homosexual couples.
Today, many gay rights advocates believe that the Supreme Court ruling paved the way for victory in the Obergefell vs. Hodges case. They maintain that when the Supreme Court rejected the legal basis behind the Texas sodomy law, it paved the way for later successes for gay rights. Meanwhile, others sided with Justice Scalia, who warned of the risk of opening the floodgates to laws permitting bigamy, incest, and bestiality. You might be interested in an article discussing the legacy of Lawrence vs. Texas.
https://caselaw.findlaw.com/us-supreme-court/539/558.html

https://thelawdictionary.org/jurisprudence/

https://www.britannica.com/topic/Lawrence-v-Texas