How did modern semicha come about

The modern semichah has never come about. The efforts of R. Yaakov Berab (Mahari Bei-Rav) to reinstitute semicha hit controversy with Rabbi Levi ibn Chabib, Chief Rabbi of Jerusalem. The Wikipedia article provides many details on the efforts to reinstitute the semicha. Another source of information is the article on R. Yaakov bei Rav in the book "The Early Acharonim" by R. Hersh Goldwrum, pp. 57-59.

As the later source states:

The ordination practiced in our times is only an affirmation of the recipient's proficiency in halacha, and is valid only for ajudicating cases for which bona-fide judge is not required.

Based on the article "What Is a Rabbi?" on chabad.org, the rabbis continued to be ordained throughout the generations in such a diminished form, since it is forbidden for a student to establish himself as an authority in Jewish law without his teacher’s explicit permission. (Rambam, Hilchot Talmud Torah 5:2-3; Shulchan Aruch, Yoreah Deiah 242:4.)

The same article notes that there is an argument on how to view the current practice.

Some are of the opinion that rabbinic ordination nowadays is a remembrance of the ancient classical semicha. Therefore they believe that when granting rabbinic ordination we should try to fulfill as many requirements of the original semicha as possible, such as the requirement that only one qualified to rule in all areas of Jewish law should be ordained (Responsa by Rabbi Moshe Isserlis 24 and Rabbi Moshe Sofer (Chasam Sofer) Even Haozer vol. 2 94.)

Most, however, believe that ordination nowadays has no connection to the original semicha. According to this opinion, there is no need to be qualified in all other areas of the law in order to receive a limited ordination. (Responsa by Rabbi Yitzchak bar Sheshet, Rivash, 271; gloss of Rabbi Moshe Isserlis to Shulcan Aruch, Yoreh Deiah 242:14; Arukh HaShulchan Yoreh Deiah 242:29. See also Responsa by Rabbi Meshulam Rothe, Kol Mevaser,1: 12 in which he notes the apparent contradiction between the two views expressed by Rabbi Moshe Isserlis (see notes 26-27) and explains that the view of Rabbi Isserlis’ that is found in the gloss of the Shulchan Aruch is his final say on the matter.) and also (Arukh HaShulchan Yoreh Deiah 242:29. See Aruch Hashulchan (ibid) where he goes further and writes that in addition, the primary function of Semicha nowadays is permission to serve as a communal Rabbi. And no one living in a city which has such an ordained Rabbi may render halachik rulings without said Rabbis consent.)

The article SEMIKHAH in the Encyclopedia Judaica notes of various reformulations and adaptations done for a document appointment to office or to acting as confirring power that has been in existence since at least 5th century. The name semikha for modern ordination reappeared in Ashkenaz (Franco-Germany) after the Black Death under the influence of diplomas and titles conferred by Christian universities. Another article "Rabbi, Rabbinate" states:

The responsa of the Sephardi Isaac b. Sheshet Perfet disclose a conflict surrounding the appointment to the office of rabbi of France in the 14th century, while other contemporary writings reveal the views of the Ashkenazi scholar Meir ha-Levi of Vienna on the nature of the rabbinate. All demand proven and attested knowledge, as well as integrity and excellence in character and conduct; and on these grounds candidates are approved and disqualified. The Ashkenazi scholar reveals a conception of a well-defined written diploma attesting to the knowledge and qualifications for a rabbi – the so-called semikhah diploma. ...

When Sephardim and Ashkenazim came in close contact after the expulsion from the Iberian Peninsula (1492–97), the institution of this diploma became a bone of contention, as is evidenced in the writings of Isaac Abrabanel. Ironically enough, the abortive attempt to resurrect the semikhah, made in 1538 by Jacob Berab and enthusiastically accepted by Joseph Caro, served to strengthen the Ashkenazi type of semikhah diploma.