Final answer:
Athens rebuilt its walls following the Second Persian Invasion, which was opposed by Sparta due to the potential benefit to Persians and Athenian power and influence. Themistocles deceived the Spartans, finishing the walls and opposing Spartan control over the Amphictyonic Congress, leading to their support for Cimon.
Step-by-step explanation:
Following the Second Persian Invasion of Greece, Athens began rebuilding her destroyed walls around the city. Sparta, arguing that this could potentially benefit the Persians in the future (although Thucydides posits that instead the Spartans were concerned about the increasing Athenian naval power and influence), opposed this. Allegedly, Themistocles then deceived the Spartans by promising to end this construction and visited Sparta himself, while simultaneously instructing the Athenians to hastily finish this work.
Ultimately, he delayed the Spartans (specifically their envoys) sufficiently for this to be achieved. Furthermore, Themistocles also greatly offended Sparta when he successfully opposed their taking control over the Amphictyonic Congress (a religious league for the running of the sanctuary at Delphi but possessing political influence). Thus, the Spartans supported his political opponent Cimon.