I can only speak to MtG when it first came out, I've been out of the loop since eh since when 4th edition started or so. MtG exploded because it was (and I hate to use this term): hackable. You could build creative exploits into the game, at our local shop there was a no CoP rule (no circle of protection) because they were considered too unfair; in hindsight this was a ridiculous rule.
Very few games embrace the unknown dynamic nature as well as MtG did. They would introduce cards that could be mixed with past-future cards almost on a whim, creating moments of pure frenzy as people managed to build systems that seemed indestructible.
To expand on Magic being "hackable": The game is this way because every rule of the game can be changed by a card. There are no set in stone rules in the sense that they will always apply the same way no matter the gamestate - there are cards that say "You can't win the game" or "End the turn", "Skip the upkeep phase", etc. The game isn't just about playing by the rules effectively, it's also about changing the rules effectively and dealing with how your opponent is doing the same. I think a related reason the game exploded is the fact that design (deckbuilding) is an integral part. Having a hackable game is great, but if the hackableness is random it's not that exciting. It's when you can design an exploit (deck) and improve on it that it gets really exciting.
To me, the main reason Magic is fun is the variety of skills and approaches it encourages. You can play it as several different resource management games: there are at least 4 important resources to manage (time, mana, cards, life) and you can win by focus on getting one or more of these resources more or by preventing your opponent from having/using any resources, and all of those decisions have major impacts on your play and deckbuilding style. You can also play it as a game of assembling your crazy exploit of the rules fast enough to win (combo decks), or you might even be able to exploit the rules so hard you're essentially playing a different game from your opponent, so you have a completely different win condition to achieve (legacy Dredge).
Yeah I tried to get back into it a couple years ago, because it is a cool game. But it was just simply no fun to play; it's obviously a game that mainly relies on being able to spend a lot of money buying power. Even at a tiny little shop, everyone had these insane combos and whatnot. Really impossible to win against without investing a lot.
Draft! 3 unopened booster packs per person. Everyone opens up a pack, picks a card, and passes to the right. Keep doing that until all the cards are gone. Then everyone opens a new deck, and passes to the left this time. Repeat for the third deck, going right. Minimum deck size is 40. This eliminates any sort of monetary advantage and privileges people who can come up with good deck combinations on the fly.
If it's just between friends, why not use self-printed cards and put some reasonable limits on number of powerful cards? The images must be available online somewhere.
I wouldn't say that's true (unless things have changed since I played). There are some decks that are dependent on many rare and expensive cards... but its also possible to build killer decks on common/uncommon cards with perhaps a few rare ones.
Deck construction really is one of the more creative aspects of MtG where you can create new kinds decks or twists on old ones
I think the allure/success of Magic is that it is many different things to many different people (and is many things to one person at the same time). WOTC describe the customers as Timmy/Spike/Johnny, but there are even more demographics in the audience. You have the gamblers who love to rip packs, the prospectors who love to buy/sell, the socialites who want to play with groups of friends, etc etc.
Very few games embrace the unknown dynamic nature as well as MtG did. They would introduce cards that could be mixed with past-future cards almost on a whim, creating moments of pure frenzy as people managed to build systems that seemed indestructible.
It's crazy to think how much some of these cards are worth now: http://www.mtgprice.com/sets/Beta/Savannah