To answer your question, it's important we look at the Athenian polis in the middle of the 5th century BCE to order contextualize Perilces' actions within socio-political and cultural growth of the Greek city-state.

After the defeat of the Persians in 479 BCE, Athens emerged as a political powerhouse in the Greek world of the 5th century. Pericles was from the wealthy and politically powerful Alkmaionidai family and in 472, the fledgling statesman--perhaps making his public entrance into the socio-political stage of Athens--financed the production of Aeschylus' new play, Persians. According to Simon Hornblower, Pericles sought to attach himself to the play as a statement in support of the politician and statesman Themistokles, who was portrayed in the play as the hero of the Battle of Salamis during the Persian War only a few years previous (The Greek World, 33-34). It's unclear whether Pericles wished to align his politics with Themistokles or whether he sought to oppose the political rise of Themistokles' opponent, Kimon, through his support of a figure more prominent than himself (or both). But, it is important to highlight that Pericles saw the theatre as an opportunity to publicly announce and promote his political agenda in Athenian society. Thus, Pericles found his efforts as a theatrones--one who leases the use of the theatre-space from the state to produce theatrical shows (Csapo, "The Men Who Built the Theatre," 88-90)--to be an effective means to support his political career and agenda.

By the 460s, Pericles had risen to prominence in the political landscape and governance of Athens. He attached himself to the reformer, Ephialtes, who supported increased democratic enfranchisement of a greater portion of the Athenian populace. After Ephialtes was assassinated in 461 Pericles took charge of realizing the vision of democratic reforms and successfully ostracized the leader of the opposing political faction, Kimon (the same Kimon mentioned previously). It was after this point that Pericles opened the theatre to the poorer, but increasingly more enfranchised, citizens of Athens. Such an act was highly political and served to identify the less wealthy citizenry within the polity of the Athenian city-state by giving them greater access to a space that was, perhaps more than others, physically encapsulated the cultural, social, and political body/bodies of Athens. David Kawalko Roselli argues that our modern divisions between ritual, politics, religious and social performance--including in the usage of space--were certainly not as distinct in the world of Classical Greece as they are today (Theatre of the People, 166-67). Indeed, theatre spaces in Athens and other Greek city states sometimes served as the official gathering point for governmental and/or other political bodies. In Athens specifically, the Ekklesia (the larger democratic body of governance in the city-state) met twice a year in the Theatre of Dionysus. In essence, Pericles was pursuing a policy that promoted inclusivity along the lines of class manifested in the space that most represented the celebrated cultural identity of Athens. Furthermore, this inclusivity was reified in the democratic reforms of his political wing that supported the growth of the Ekklesia.

To sum up, Pericles saw opportunity to gain popular support for his political reforms by bringing more Athenian citizens into the audience attending theatrical productions at the Theatre of Dionysus. Pericles not only began his political career through efforts in Athenian theatre but maintained this particular practice throughout his political career. His named is attached to the Odeon of Pericles--a structure that was utilized for gatherings surrounding the Dionysian festival and theatrical production--during his tenure as the leading statesman of Athens. He not only transformed the political landscape of Athens during his life, but the cultural landscape as well.

Bibliography:

Simon Hornblower, The Greek World 479-323 BC, revised edition (New York: Routledge, 1991).

Maarit Kaimio, "The Citizenship of the Theatre-Makers in Athens," in Würzburger Jahrbücher für die Altertumswissenschaft 23 (1999): 43-61.

David Kawalko Roselli, Theater of the People: Spectators and Society in Ancient Athens (Austin: University of Texas Press, 2011).

S.C. Todd, Law, "Theatre, Rhetoric and Democracy in Classical Athens," in European Review of History--Revue européenne d'histoire 12.1 (March 2005): 63-79.

Sebastian Trainor, "The Odeon of Pericles: A Tale of the First Athenian Music Hall, the Second Persian Invasion of Greece, Theatre Space in Fifth Century bce Athens, and the Artifacts of an Empire," in Theatre Symposium 24 (2016): 21-40.

Michael Vickers, Pericles on Stage: Political Comedy in Aristophanes' Early Plays (Austin: University of Texas Press, 1997).

Peter Wilson, editor, The Greek Theatre and Festivals: Documentary Studies (Oxford: Oxford University Press, 2007).

EDIT: Fixed some syntax and adding a short clarification about Klimon.