First of all, my guess is that it is the overall challenge level of the games that are most frequently played at these sessions. Unlike the board games I was familiar with as a child, Monopoly, Life, and even Scrabble, these games are generally much more complex. It often takes almost a quarter hour just to explain/teach many of the games to new players.
Secondly, the best of the games require flexibility with strategies. There is usually either an element of chance in the game or a factor of randomness that insures that the strategy used for one time play of the game may have to be completely changed for the next time the game is played.
One of my favorite "board" games is Dominion. It probably shouldn't be called a board game, because it is really a deck-building game, but it is a frequently played game at these MeetUps. The general structure of the game is that, on your turn, you first perform any actions that your cards allow you to perform (A), then you buy any new card(s) you are able to buy with the cards in your hand (B), and then you clean up your play area and draw a new hand from your accumulated cards (C). Some of the cards you can buy add actions you can perform, some give you more money to spend, some only win you victory points at the end. The action cards are where the vast differences in games come in. There are around 30 cards in one particular version of the game. Of those 30, only 10 are selected each time for play. This makes for a huge number of combinations.
Play also depends on the personalities and strategies of your opponents in the game. Some of the action cards are attack cards, which have a huge affect on how the game plays out. Aggressive players can wield them to great advantage; more peaceful players (like me) can get taken by them. Some cards have cumulative effects - one card allows you to play another, which may allow you to play yet another. Clever use of these cards can be very effective.
An Aussie, SH, introduced me to role-playing games, a Harry Potter one, about 4 years ago and it was great fun. RPG games add another dimension to gaming, akin to acting. The emphasis here shifts to solving a problem with other players (or against other players), as the character you are playing would. The MeetUp group I go to also occasionally plays a semi-role-playing game called Werewolf. Roles are secret and you must discover who is a Werewolf, before everyone in the village is killed. It is more constrained than regular RPGs and follows a similar format every time, but the complexity comes from the interactions with others in the game.
What is the appeal of gaming? It requires personal engagement with the game and with other people. There are rules to the games, but there are many different winning strategies. There are a lot of other types of play that gifted people like as well, but I think the active involvement with challenging and varying games is a big plus for gaming. I came to this kind of gaming quite late in life - I am usually the oldest person in the room for these games - but I have enjoyed learning them.