How the Secret of this Killing became known
Allah tells us that when Musa killed that Coptic,
فِى الْمَدِينَةِ خَآئِفاً
(he became afraid in the city) meaning, of the consequences of his action,
يَتَرَقَّبُ
(looking about) means, turning around and watching out, waiting for the consequences of his action to befall him. He went out and about, and saw the man who sought his help the day before, fighting with another Coptic. When Musa passed by him, he called for his help again, against this other Coptic. Musa said to him:
إِنَّكَ لَغَوِىٌّ مُّبِينٌ
(Verily, you are a plain misleader!) meaning, `you obviously lead people astray and are very evil.' Then Musa intended to attack that Coptic, but the Israelite -- because of his own cowardice and weakness -- thought that Musa wanted to hit him because of what he had said, so he said, in self-defence --
يمُوسَى أَتُرِيدُ أَن تَقْتُلَنِى كَمَا قَتَلْتَ نَفْساً بِالاٌّمْسِ
(O Musa! Is it your intention to kill me as you killed a man yesterday) Nobody except him and Musa, peace be upon him, knew about it, but when the other Coptic heard this, he took the news to Fir`awn's gate and told him about it. So Fir`awn came to know of it, and he became very angry and resolved to kill Musa, so he sent people after him to bring him to him.
Tafsir Ibn Kathir (abridged, English).