This is a difficult question to answer, simply because everyone's definition of a "good public speaker" is going to be slightly different. So I want to start off by saying that what follows is more specific to Hitler himself and not necessarily a blueprint for public speakers writ large (although someone wanting to become a good public speaker could do worse than follow Hitler's blueprint). It's also worth noting that, as far as I know, the main measure that the Nazis used to determine a particular speakers proficiency were gate receipts. The Nazis almost always charged admission to their events, and so it was concluded (not without reason) that those speakers who sold the most tickets were the "best". So when we say that Hitler was an excellent public speaker, what we really mean is that he was able to sell a lot of tickets to the events that he spoke at, not that he was somehow uniquely persuasive in his speaking or some other, less quantifiable measure.

Hitler was successful in public speaking because of various factors all of which combined to make him very effective.

1. He spoke clearly, despite his yelling and theatrics. This is difficult to "prove" in a traditional sense without a direct analysis of the language used and the recordings themselves. But if you have any decent facility with the German language, you can go to Youtube and listen to some of his speeches. Odds are, if you're half decent at German, you'll be able to understand him, even if German is a second language to you. In this period before TV and when radio was in its infancy, it was important to be able to project your voice (loud speakers were not always present) to large audiences clearly. Hitler did this very well. He was able to get his points across to rooms that held hundreds, or even a few thousand, people. This is no small feat. Give one of his speeches a listen here. He's screaming, but if you can speak German, despite the poor quality of the recording, you're likely able to get the basic message.

2. What we view as crazy theatrics, is really a seasoned veteran speaker working over his audience. These famous pictures of Hitler rehearsing a speech in front of a mirror in 1925 are a perfect example of this strategy. As with clear speaking, these sorts of theatrics helped the audience understand the significance of certain points he was trying to make, because not all communication is verbal. Two examples of how this works: First would be silent films, in which movements and facial expressions are over-exaggerated in order to elicit a reaction from the audience. (remember that the first talkie film was only released in 1927 and they didn't take worldwide until the 1930s). And more currently, think of the way that professional wrestlers speak to their audience, especially in live performances. A certain amount of over-the-top theatrics are expected and rewarded through enthusiastic responses by the audience.

3. Hitler and the Nazis knew their audience and tried to be all things to all people. The Nazis were famously negative politicians. Yes, they had a program, but in their campaigning, they almost always focused on the negative - that is to say, they focused on what was going wrong, not on how they were going to fix it. This is a strategy used by outsiders (of which the Nazis certainly were in the early and mid 1920s) or opposition parties against the governing party in many countries - the US included, and its time tested and mother approved, because it works marvels. Most of us are probably familiar with many of the outlines of this message:

Defeat in the war (World War I) was because of a stab in the back by a Judeo-Bolshevik cabal working to overthrow European civilization. Germany was never defeated on the field of battle and the Versailles Dictation was forced upon the up-right German people by foreign powers and a weak, overwhelmingly Jewish, 5th column. Democracy and crony capitalism, along with Judeo-Bolshevism, were the prime enemies of the German people desiring to split the national community along economic lines instead of creating a harmonious community of happy German super-humans along racial lines. And, of course, Hitler and Hitler alone, would set it all right. Hitler would bring honor and glory back to Germany. And so on and so forth.

This sort of message - one of modern decay and a promise to return society to a better, golden age - is very popular, regardless of when and where. Yes, this particular version has German specifics in it, but the general theme is one used time and again by politicians of various stripes in various locations. I would be remiss if I didn't point out that recent American politicians have made promises that are - at least in the general theme - very similar to the ones the Nazis consistently made. I'm not trying to equate the two - that's a mistake - but simply point out the usefulness of such a strategy for winning over an audience and to gain political support.

4. Hitler's appeals were based on emotion, not facts, evidence, or sound logic. Politicians have long known that the way to engender loyalty to a movement it to speak to the audience's emotion - people who feel are more reliable voters than people who think (What a depressing thought!). Must of what I discussed in Point 3 are appeals to emotion, but the importance of this merits it own point. Whether scapegoating the other or building up the purported importance of the German Volk, Hitler was playing a game of emotional manipulation with his audience. He was telling his audience that they were great, worthy of a better life, and that their trials and tribulations were not of their own making, but all the fault of someone else, anyone else, really. Important to this distinction is that the folks that were the "real Germany" were the ones in attendance. It was those who were absent that were stupid for not seeing the truth that those in attendance so clearly understood. In this way, he made his audience into a sort of vanguard for the eventual brown revolution that would correct society's ills.

5. Hitler practiced sound editing. I tell my students all the time that the key to good writing is revision, revision, revision. No one writes a masterpiece on the first try. Hitler knew this and practiced this method. Goebbels, in his diary, notes that Hitler was constantly revising his speeches, sometimes as many as five times before he would give it publicly. These constant revisions make for a better finished product, whether it's with undergraduate writing or national political speeches. To get the best product, you have to put in the time to make it good, and by all accounts, Hitler did just that. Ian Kershaw also notes that Hitler often worked into the night revising drafts he had made earlier that day.

6. The largest rallies, such as the annual party congress, usually held in Nuremberg, were the mega events of their day. With our TVs and our Internet, we 21st century folks often lose track of the way the world was before these inventions. When you couldn't stare at a screen for hours on end, you had to find ways to amuse yourself and going to large rallies, parades, and the like were easy ways to spend an afternoon. To this day, many folks love going to these sorts of events (think of the Opening Ceremony of the Olympics or the Macy's Thanksgiving Day Parade) to see all of the entertainers perform their acts. There was a similar thing going on in Nazi Germany, in which tens of thousands, or more, would show up to see the SA marching in formation, or to watch the local HJ or BDM group participate, or to see a speech given by some local or national political figure. The popularity of such events is evidenced, not least by the huge crowds who showed up to them, but also by the popularity of Leni Riefenstahl's Triumph of the Will, which was a propaganda film that used many cinematographic innovations to get across its message. It was, more or less, a chronicle of the 1934 Nuremberg Rally, and when it was released in 1935 in Berlin, it became an instant success. People wanted to see these events, even second hand on film a year later, because they were major, major productions. Even if you didn't like the man or the party, seeing one of these spectacles was something worth seeing for the very sake of it, and so Germans went.

These are some of the reasons that Hitler was successful. If you're interested in learning more, I'd suggest various works:
Victor Klemperer's Language of the Third Reich goes into a lot of detail about how the Nazis used language in propaganda.
Peter Fritzsche's Germans into Nazis discusses the electoral system in the Weimar Republic and discusses the types of people that would consider voting for the Nazis.
Ian Kershaw's Hitler: 1889-1936 Hubris discusses his speech writing and speaking.

As always, Evan's Third Reich Trilogy, especially the first two volumes, are a great source for all things Nazi related.

If you're asking about his speaking style specifically, then Goebbels has some words to say about this. In his 1936 book, Goebbels talks about 2 different types of speakers (those that speak from the Heart and those who speak from Reason). He says that the Fuhrer is able to reach out to both sides of the audience and drive them towards his own position.

"The Führer is at his best, however, before a small audience. Here he is able to reach each individual member of the audience. His speaking carries away the listener, who never loses interest because he always feels spoken to directly. He may speak about a random theme with an expertise that astonishes the specialists, or in speaking about everyday matters suddenly raise them to universal significance."

Hitler would often arrive late to his own speeches in order to drive up the tension and the atmosphere of the crowds. He would start off as being soft spoken but then he would erupt into a cascade of passion and fiery discourse. After this discourse, he would gradually settle down again and promptly leave.

Source: Joseph Goebbels, “Der Führer als Redner,” Adolf Hitler. Bilder aus dem Leben des Führers (Hamburg: Cigaretten/Bilderdienst Hamburg/Bahrenfeld, 1936, pp. 27-34.

Whilst you wait for some good answers, most of this is covered in the Wiki. This post by /u/OnkelEmil is close to what you're asking about but your secondary question about his particular speaking style is interesting and I can't recall it ever being asked here before.

(edit was because of a spelling error).

https://www.reddit.com/r/AskHistorians/comments/5bs2hk/hitler_won_control_of_the_nazi_party_with_an/d9r5a2a/

This is a response by /u/boffcheese that addresses the first part of your post I think.