At its simplest, analysis is “telling the judges why what you say is true.”
The opposite is an assertion. So if I say that “If we do this people will be happier” that's an assertion. If I say “When we do this it will cause A, A will cause B because of C, […...] and then X will happen which will make people happy because Z.” See what I mean?
Metaphors I find useful are building links in a chain, or piling bricks on top of each other. A lego house made of small bricks connected together and reinforcing one another is going to be stronger than a loose pile.
Saying more words, or saying longer words, isn't the same thing as analysis.
Explaining something in a roundabout rambling way that doesn't actually include the important content of what you are saying. Just because you've talked about something for four minutes doesn't meant you have actually explained what it is in any level of detail. "A then B then C" is different from "A! C! !B Z! A!"
Long and technical words may make you sound clever, but they don't actually make what you are saying any better unless they are explained or bring something extra with them. "The economically disenfranchised are systematically excluded from the political discourse" doesn't actually tell me anything more than "poor people don't have much power in politics."
Strategically deploying analysis
Now you obviously don't need to do a huge amount of analysis for every claim you make, you can probably get away with saying “water is wet” or “death is bad” without much explanation. And in the limited amount of time you have to speak you need to strategically decide which of your points need to be analysed more. So you focus your time on points the other side are likely to oppose.
So your opponent will be more likely to question a point that is important to the debate (e.g. free speech is an inherent right). This also means that a point that doesn't need analysis in some debates will need a lot in others, so in a debate on military intervention people are unlikely to question that death is bad, but in one on bioethics they might say it either isn't bad in itself or less bad than other things.
On the flip side, if I spend half my speech proving in detail that "Democracy is good" and my opponent doesn't contest that, but contests how well my method achieves democracy then my analysis won't really help me win.
With any point the two things you need to prove to the judges are:
1) Why is this true
2) Why is this important.
(Though admittedly the boundary between them is often fuzzy).
We've already talked about proving why something is true, and proving why it is important works the same way. Often its easier to prove importance than truth since its often more intuitively obvious, ("X will cause a civil war" is a difficult thing to prove, "civil war is bad" is easier). You may also have to prove why something is more important than another thing, which is normally called "Comparative" but that needs a separate article.
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