Why did Plato hate democracy??
There are two separate issues here, Plato's specific criticisms of the Athenian democracy and Plato's attitude towards democracy as a system.
Plato was far from the only person to have concerns about the Athenian democracy. Many other writers portray the demos as being easily swayed or even bribed by demagogues, and prone to factional strife (stasis), violence, irrationality, and a sort of moral blindness. It was the democratic faction in Athenian politics that supported the Peloponnesian wars, senseless conflicts that resulted in a huge death toll and the eventual fall of Greece to Macedonia. The democratic faction also wanted to prevent the Melians and Mytilenians from remaining neutral in the wars by enslaving all the women and children and killing the men. Perhaps most importantly, it was the Athenian democracy that killed Socrates.
On a more philosophical level, Plato was concerned that democracy stood in opposition to expertise. Just as one would want a skilled doctor rather than a randomly selected citizen to perform brain surgery, so Plato felt that in matters of laws, city planning, administration, and so on, the city was better off being run by experts. He also was concerned about corruption and thus in the Republic argues that the Guardians should not be allowed to own private property at all so that they will not be swayed by personal gain but will always consider what is best for the city.
According to Plato, a democracy could turn into an anarchy in which every citizen is out for himself and no one has the long-term needs of the state in mind when making political decisions. Plato wanted a nation to be governed by elites who would look after the needs of all the people in a way that ensured that the state continued to thrive and prosper. Plato's focus on "order" over "freedom" has been taken in other republics as well. The Founding Fathers feared what would happen if politics became a popularity contest. Plato's ideal leader might be unpopular, but he would at least be wise. It is unlikely that the people would choose the wisest person to be king; rather, they would like the person who would give them everything they wanted.
Well, "hate" may be too strong of a word, but he certainly wasn't a fan. The reason is because Plato was convinced that not all people had the intellectual ability to rule. It is an elitist position, to be sure, but Plato argues that the masses are not capable of deep thought and need a select number of "philosopher kings" to guide them.
A democracy, by definition, is rule by the people. Rarely are those chosen by "the people" philosopher kings...