In neither mythology or bronze age Greece there was a empire. Mycenea was basically a first among equals, respected and followed in times of turmoil but could not enforce anything as if it was a imperial capital.
There is two things that made Agamemnon the "leader'. First is that he ruled Mycenea. Mycenea is among the oldest cities (altrough it is constantly confused with Argos) of the Peloponnese, and since the earlier kings like Phoroneus and Argos they had sway over Corinth and Sicyon as well. Later, king Abas extended the city influence up to Iolcus and Phthia. Abas was the father of king Acrisius, father of Danae, mother of Perseus with Zeus. After Acrisius' accident, Perseus ruled as a Anax from Mycenea with Andromeda by his side.
Perseus' son Electryon became the king and entered into a war against the Teleboans. His nephew Amphytrion defeated the Teleboans and conquered the islands, awarding them to Cephalus, who becomes the father of Arcesius the builder of Ithaca. These islands were now ""under"" Mycenea as well. Amphytrion killed his uncle the king by accident (or not) so he fleed to Thebes with his wife Alcmena, daughter of Electryon. Because Electryon's sons died in the aforementioned battle, Sthenelus, the other son of Perseus, became king and the father of the next king, Eurystheus, while Alcmena exiled in Thebes became the mother of Heracles with Zeus.
While Heracles served Eurystheus, he conquered a bunch of cities. Heracles conquered Elis from king Augeas, and Augeas' son Agastenes became king but now as a ally of Mycenea. Heracles then conquered Pylus from Neleus son of Poseidon, who had refused to purify Heracles from murdering Iphitus, and Heracles even defeated Hades and Poseidon in the city as well. He put Nestor, Neleus' son, in the throne, now as a ally of Mycenea. Finally, he conquered Sparta from the usurper Hippocon, and gave the city to the true king Tyndareus, the father of Castor, Pollux, Clytemnestra and Helen (but Zeus was the true father of some of these). So Tyndareus had a debt to Mycenea now.
After Heracles became a god, Eurystheus wanted to kill Heracles dozens of sons as they were rivals of his family. But the Heraclidae took refuge in Athens, so there was a battle and Eurystheus and his sons were slain. Before or later than this (the timeline is contradictory if we combine different authors together) Castor and Pollux conquered Athens in the name of Sparta to rescue their sister Helen, and they put Menestheus in the throne of Athens as a ally.
After the end of the House of Perseus, Eurystheus maternal uncles Atreus and Thyestes became the kings of Mycenea, and they were the sons of king Pelops, son of Tantalus (that one). The brothers story I will not explain here, but Agamemnon and Menelaus, sons of Atreus, had to flee to Sparta where, with support of Tyndareus, they re-conquered Mycenea from their uncle Thyestes. Agamemnon became king and married Clytemnestra, Tyndareus' daughter, in a political marriage. Menelaus married Helen, the other "daughter" of Tyndareus, and after Helen fleed with Paris, and Castor and Pollux died in the same moment in a battle elsewhere, Menelaus became king of Sparta because he was married to Helen.
As you can see, Mycenea and its ally Sparta through the mythology subdued several cities. But these cities were not forced to follow Mycenea or pay tribute, is just that problematic kings were removed in favor of their less dangerous family members that would now go to war allied with Mycenea instead of against Mycenea and Sparta. Agamemnon simply inherited most of that after defeating his uncle.
The second thing that gave power to Agamemnon was his scepter. He inherited the scepter from his defeated uncle Thyestes, who took it from Atreus, who had received it from his father Pelops. Pelops had received it from Hermes who received it from Zeus, and it was forged by Hephaestus. So this scepter gave Agamemnon divine authority. So when Paris brought Helen with him, the greek kings had to honor their Oath to Helen as well as the fact they wanted to sack Troy, and since Agamemnon had the most amount of allies and the divine Scepter, they elected him as their overall ruler until the war was over. But he could be replaced as well: In certain versions of the Iphigenia myth, Agamemnon refused to kill his daughter, so the other kings threatened to kill him and replace him with Palamedes, as Palamedes was intelligent, cold and calculating, so the perfect ruler for them. But Agamemnon does go with the sacrifice and is not replaced.
I have seen many people insist that Agamemnon had the ambition to conquer Troy and he kinda forced everyone. When it can be argued that is the opposite. The kings elected him, and in the Iphigenia episode, they were forcing Agamemnon to take a decision. Agamemnon was not automatically the ruler, but the most powerful king and the one with divine aproval from Zeus, so the kings choose him to fulfill their ambition: the Oath they had to fullfil to Helen and their desire to conquer Troy and everything it had. Everyone was power hungry, not just Agamemnon, instead of blaming only him, he was instead a manifestation of the Achaean greed (this is why most of them suffered later, not just Agamemnon). He was also less cold-hearted than the likes of Palamedes, and his main reason to go to war was to defend the name of his brother that was under shame, so these are noble reasons. As for his flaws, well, after Aphrodite rescued Paris from the battlefield nullyfing any possibility of peace, Agamemnon said to his brother to not show any mercy to any Trojan as they were not trustworthy, so only from that point onwards he became excessively cruel. His biggest flaw was certainly everything sorrounding his wife, as he hated her and only married for political reasons, and this is what lead to his downfall in her hand and he didn't enjoy any peace after Troy, but died imediatly.