Micro-transactions are small purchases that can be made with in a game, usually for in game items as opposed to being a full expansion. Often micro-transactions can include in-game currencies, skins, power-ups and equipment such as armour, weapons and vehicles. Companies often use these to continue funding a game after all or most sales of the game itself have been made with little effort on their part to create new content, with many companies being accused of cutting content originally intended to be apart of the full game so it can be sold later as a micro transaction.
Much of the controversy surrounding micro-transactions comes from the general displeasure of having to pay for content someone should already have access to but that is still relatively harmless regardless of how dishonest it is, however micro-transactivisions can have a great effect on gameplay.
Several multiplayer games have tiered systems of weapons, equipment,etc. that can be exploited by micro-transactions or are even sometimes designed around micro-transactions. The reason this is a problem is that the 'best' players of these games will no longer be the most practised and skilled players but instead it will be some asshole who looks suspiciously like Casper the friendly ghost

Micro-transactions are the idea of a money hungry fat cats trying to scrape every bit of cash the can from their consumer and hoping they are dumb enough to fall for it, are you? Well yeah probably, I mean it's still happening. Look just do your best not to fall for this shit and maybe they'll stop, and maybe never download a game on your phone, like ever, just watch Southpark.