(This is a very basic run down of what I understand to occur. There are places where I omit steps and oversimplify, but I THINK it's pretty accurate. Someone please correct me if I'm wrong.)
Mining is basically the way that everyone agrees a transaction is valid - you have a bunch of unrelated entities saying, "Yes, according to my copy of the ledger, the sending party has X dollars and wants to send Y dollars, and the two parties agreed." (It's much more nuanced, but this is the easiest way to summarize it.)
If someone said, "I'm going to send AccountB 100ETC from AccountA," but AccountA only had 50ETC then a miner would see that and deny the transaction (wouldn't add it as a valid block to the blockchain). In reality, there are several places that the transaction should be thrown out, but there are obvious ways around those, and every transaction incurs a fee, so trying to just flood the network is costly. More importantly, several miners have to agree that the transaction is valid. When sending bitcoin, you'll see a number of "confirmations," which is the number of miners that marked the transaction as valid. You can see this when sending coins between two exchanges, you'll see that the receiving exchange won't allow you to use the coins for trading until you hit, say, 30 confirmations.
The problem with this model for smaller projects is that those "confirmations" by unrelated miners can't be trusted if one person/group owns more than half the miners. I left out the idea of wallets/nodes to bring this up here. Every "full" wallet (or node) has a copy of the blockchain and does some cursory checks on transactions before broadcasting them to the network. Miners then package a bunch of transactions in a block and send this to the network - "Here is my version of block X. It should be added to the blockchain as block X for everyone." The miners then have to expend some effort to verify that the block is actually the block they think it is and contains valid transactions. When the miners confirm a block, it eventually propagates to all the miners and nodes, and competing blocks by other miners, which might have contained some of the same transactions are discarded. It doesn't mean the other blocks were fake, just that the chosen block reached "consensus" of the miners and can be trusted to be valid. (I realize this leaves lots of open questions, but I'm limiting the scope to OP's question.)
Consensus is the big problem in 51% attacks. If someone controls more than half the mining power, they can reach consensus on blocks faster than legit miners and add whatever they want to the blockchain that is distributed to all the other miners and nodes. That means they can add/delete transactions or manipulate existing ones to, say, change the receiving address to their own, and everyone that receives those blocks will accept the outcome.
This is why decentralization is so important to cryptocurrency. There's a level of necessary chaos that keeps the network honest. It is possible to roll back the blockchain to a previous version, but that also means that transactions in all blocks that followed must be rolled back as well. If the rogue party simply accumulated currency, that's not a big deal, since all other transactions would still be valid. But, if they immediately turned around and traded them for other coins, you start to see the problem. On top of that, if ETC was more "robust", the news would (and currently is) send the price into a nosedive, which creates an opportunity to legitimately buy it very low and profit from the recovery. That's not going to be the case here.
All this being said, Ethereum Classic was kinda declared dead a month ago when the development group halted operations, so anyone with any significant amount of ETC should have sold off by now. This attack will probably be the final nail in the coffin.
TL;DR - In the end, the state of the blockchain comes down to a simple majority of miners agreeing that a block (a group of transactions) is valid. At 51%, you can force everyone to see invalid transactions as valid. ETC development ceased around a month ago, so no one that follows it should be holding any now, but people blindly trading for profit will get a nasty surprise.