This is a reasonable start:
https://en.bitcoin.it/wiki/Block_size_limit_controversy
First, most people want Bitcoin to scale such that more people can use it. This can be done directly on the blockchain by allowing more transactions to be recorded in each block. Or it could be done with '2nd layer' solutions that periodically resolve on the Bitcoin blockchain. Such solutions require some changes to the Bitcoin clients people operate; the current 'most popular' version of this is called Segregated Witness, aka SegWit.
Right now, more people want to create transactions than can actually do so, thus a fee market has formed to get transactions included in blocks. This is why fees have increased.
It's a debate that sounds like a technical detail, but in reality it goes straight to the core of what Bitcoin is and how it will work. There is the debate over what changes will be made, but there is also a debate about how this decision will be made. Some feel that all changes to the Bitcoin protocol should be made with strong consensus, a conservative approach that has worked alright up to this point. Some still hope that this can be done, but the deep ideological division that belies this debate makes that seem unlikely. The alternative view is that Bitcoin is designed to handle these disagreements naturally by hard fork.
A hard fork is a split of the Bitcoin blockchain into different incompatible chains. In this case, it would be a split over the maximum permissible block size; one group would allow larger blocks, and another would refuse them. This would lead to a situation where miners choose which chain to mine on. Miners are incentivized by mining rewards (coinbase rewards), and thus have an economic incentive to mine on the chain they believe will have the greatest market value. This will likely result in a positive-feedback mechanism where one chain dominates the other; this has played out on the Ethereum chain where 'classic' ether is worth far less. A stable split is also possible, and thus there would be two versions of Bitcoin with different rules. Any coins generated or transacted pre-fork would be valid on both chains, but post-fork would commit to one chain or the other.
As for the bigger blocks, it comes down to predicting how the network will behave with larger blocks. It would permit more transactions, without question. It would also increase the bandwidth and data storage requirements of full-node operators. Basically, it would be more expensive to operate the Bitcoin network (which is distinct from the cost to secure it, which is scheduled coinbase inflation and txn fees). It would also mean that miners that are closer to each other on the network (via latency and throughput) would have a competitive advantage as they would confirm each other's valid blocks more quickly, and thus there are concerns about miner centralization. Such centralization presents a point of failure for the network.
Make no mistake; it is very difficult to predict what will happen with different protocol parameters. There's a lot of game theory that you can do to try to understand it. For instance, miners may mine larger blocks to include more transactions, and thus transaction fees, but risk slower propagation of their valid block through the network. In the case of valid blocks being found at the same moment, the one that propagates faster will 'win' and the other will be orphaned. Some argue that the risk of orphaned blocks is an effective limit to blocksize, and thus no blocksize limit is needed. It would also peg transaction fees to the marginal risk of a block being orphaned.