To justify is to make excuses for one's actions, like the dieter who has to justify the few tiny pieces of chocolate she's sneaked in between salads.
Anyone who has ever made excuses for his or her actions knows the meaning of justify, like the bank robber who justifies his latest heist by claiming he needs the bank's money to support his charitable foundation. He might think he's got a good reason for holding up the bank, but he's going to have a hard time convincing the bank's owner of his reason, or justification. To justify a line of text is to space it so that it lines up with the margin.
To discuss is to talk: but with a more serious intent. Discussing is a lot more substantial than chatting.
When someone says, "I have something I need to discuss with you," most people think "Uh oh." To discuss usually implies a certain amount of serious talking. This can apply to writing as well: essays discuss topics, usually in a careful, thorough way. On the other hand, sometimes discuss means exactly the same thing as talk or converse. Like all words, the meaning depends on how it’s used.
To explain something is to define it, show how it works, or just tell what it is. Explaining helps people understand.
If you know football really well, then you could probably explain it to a non-fan. Explaining makes things clearer. A lot of teaching is explaining — telling how something works. Also, if you do something wrong, people will ask you to explain your actions. Explaining can help someone "get it" — whatever "it" is.