Štilec was born in Prague on July 18, 1985. His family was musical, and he enrolled at the Prague Conservatory as a violin student. Switching to conducting, he studied with Oliver von Dohnányi and Leos Svárovský at the Prague Academy of the Performing Arts. At the age of 17, he founded his own ensemble, the Quattro Chamber Orchestra. A breakthrough came in 2010 in the form of an invitation from Tilson Thomas to study with him under the San Francisco Symphony's auspices; that led to a stint under Tilson Thomas' supervision conducting the New World Symphony in 2014. Štilec has conducted various orchestras in the Czech Republic and elsewhere, including the Prague Philharmonic, the Czech National Symphony Orchestra, and the Czech Radio Symphony Orchestra. Abroad he has conducted the Ulster Orchestra, the Orchestra of the Swan, and the London Classical Soloists, among others.

It is with the Czech National Symphony Orchestra that Štilec's name has been most widely distributed on recordings. He was contracted by the Naxos label to lead the orchestra in a complete cycle of the orchestral music of composer Zdenek Fibich. When the cycle began in 2013, Fibich was little known beyond a group of highly descriptive romantic piano pieces. The five-CD cycle concluded in 2014 with Fibich's overtures and ballet music from Hedy, and it immediately led to more recording opportunities for Štilec. In 2017, he and the Czech Chamber Orchestra Pardubice recorded symphonic music of composer Leopold Koželuch, and the following year, he joined with the Prague Symphony Orchestra and the Martinu Voices to record that composer's Cantata for the Coronation of Leopold II, the same event for which Mozart's opera La clemenza di Tito and Piano Concerto No. 26 in D major, K. 537 ("Coronation") were composed. Štilec, who by then had established a reputation as a conductor who could move at will in the top echelons of Czech music, has also led the Hradec Králové Philharmonic Orchestra in recordings for the Arco Diva label. He has maintained a vigorous release schedule on Naxos, continuing to issue music by Koželuch -- not only symphonies but also the Masonic cantata Joseph der Menschheit Segan. The years 2020 and 2021 saw Štilec begin new cycles of music by little-known composers: Viteslav Novák: Orchestral Works, Vol. 1, featured the Moravian Philharmonic and was followed by Paul Wranitsky: Orchestral Works, Vol. 1, with the Czech Chamber Orchestra Pardubice. ~ James Manheim, Rovi