Crosswords, like art, humour, and xml, should be able to stand on their own as self-contained units not requiring further explanation. If it doesn't make sense unless you explain it, you're doing it wrong.
That said, to fully appreciate any of the above, you have to understand the conventions. A knock-knock joke won't make sense if you've never used a door.
In the interest of making my work approachable, I'll try to demystify the solving process. Constructors and editors adhere to the following rules and conventions in order to make the game fair for the solver.
1. Answers are never contrived - they are always real people, things, places, turns of phrase, etc. You may not know them, but we haven't invented them.
EXCEPTION TO RULE #1: Many puzzles have themes. A themed puzzle has several entries that share something in common (I'll go into theme types later). Often, the theme entries are novel phrases - HOWEVER, they will always be formed by applying a transformation to a real phrase. In other words, the only time we invent grid entries is when the act of invention itself is the theme.
2. Clues and answers always agree in tense, gender, plurality, case, etc., and generally the "substitution rule" will always hold. This means that the clue is always perfectly synonymous with the answer, to the point that it one could be substituted for the other without creating grammatical or logical inconsistency. For example, [Beasts] clues ANIMALS while [Beast] clues ANIMAL, [Took on] clues HIRED while [Takes on] clues HIRES, etc.
3. Along those lines, there are several ways in which the form or style of the clue tells you the form or style of the expected answer. Obvious (i.e. non-idiomatic) foreign words or names, as well as foreign place names, mean that the answer is a foreign word ([Summer in Paris] clues ETE, ["Hi, Jose!"] clues HOLA, [2007, por ejemplo] clues ANO, etc). If the entire clue is a phrase between quotation marks, then the answer is a synonymous phrase (["Enough already!!"] clues CANIT, or SHUTUP, or LETITGO, or something like that). Obvious abbreviations and acronyms imply that the answer is also an abbreviation or acronym. This most often pops up with answers like IRS, FBI, and NRA clued with things like org. or assoc. in the clue. EXCEPTION: the use of e.g. is exempt from this rule. Example type clues are common enough, and print space at enough of a premium, that e.g. is used WITHOUT denoting an abbreviated answer. (Side note: a good crossword will have very few abbreviations, and no uncommon ones. If your gut tells you that a clue like [Belgian zoological org.] for something like IOEBEK is unfair, you should probably take heed and run the other way).
4. Clues ending with a question mark denote one of 2 types of answers. If it's a non-theme entry (or if not every theme entry has one) then the ? denotes trickery, puns, and wordplay (more than usual). These are often the most fun and rewarding clues, once you get them. If every theme entry ends with a ?, then there's a very good chance that the answers are invented phrases, and the theme is the transformation that's been applied to arrive at each novel answer (or sometimes puns or riddles about a common subject). For example, if the theme is "change a short e sound to a long e sound," you might find ["Mr. Cowell, grab that 'American Idol' contestant!"?], which clues SIMONSEIZE, and [Smart fowl?] which clues EDUCATEDGEESE (from Brendan Emmett Quigley's 10-21-2007 NYT puzzle). Get it?
THEMES
Remember what I said above about things being self-explanatory? This is where it really counts. A puzzle's theme is sort of like an additional mini-puzzle within the puzzle, and, like any good puzzle, should be hard to see when you start, and impossible to not see once you've solved it (I, of course, will likely miss the mark often in my own construction, but I try my best). However, you still have to have an idea about what to expect. Theme entries are sometimes identified in the clues and sometimes not, but are easy to identify because they are almost always the longest answers in the grid (and more often than not they all run ACROSS). Since there are so many types of themes (and constructors are always toiling away trying to come up with novel ones), and because of what I said above about themes ideally being self-explanatory, I won't say much on theme types. Briefly, common themes include: a word or string of letters contained within each theme entry, a single transformation applied to a number of common phrases to form wacky new phrases (which are clued with ?s), phrases that each begin with a word that can precede/follow a single other word (which will be included in the grid with a clue explicitly tying it to the theme), a series of puns about a single subject, a quote or quip broken up over several answers, and the diabolical rebus themes where several letters, symbols, colours, or even entire words have to be entered in certain squares (of course, you are not told this off the bat).
Whew! I had hoped to write a few lines about this, but I ended up not even scratching the surface in a few pages. Trust me, it's not as complicated as it sounds, and all of these rules are really there to make solving fair, enjoyable, and natural - once you get the hang of it by just solving, you'll never have to think about the rules (except when they're broken, for better or worse). If you do want more (MUCH more info), check out BarelyBad's reasonably exhaustive dissection of the cruciverbal organism here. Cruciverb.com is a great resource for constructors, and they have a decent list of theme types here.
Lastly, hit me up if any/none of this makes sense and you want to get into solving. I'll be gentle, I promise!
