The EIP 1559 paper, available on GitHub, introduces a new type of transaction that does a number of things. It makes some claims, but they are dubious. Most of the text covers the intent of the changes which does not align with the actual impact it will have.
tl;dr - adds a neat way to keep prices of using the Etherium network low but throws in an unrelated tax on the poor and actual users to benefit the rich early adopters who hoard their tokens.
Most of the changes are unrelated, but just bundled into the new transaction method. If it were a code commit I'd have requested they break it up into different parts. It's worse than a government spending bill after the lobbyists had their way.
Transactions are processed by miners and the miners are rewarded like in most other distributed finance or 'bitcoin' systems with some coin for doing that work. This includes 'gas' or a direct fee to the person who requested the transactions. This proposal changes this gas system.
Some of the features are good such as discounting gas to encourage miners to set lower costs on processing and penalizing higher than average gas costs. The replaces the free market that drives the cost up to match how hard it is to compute the transaction.
The EIP also requires gas be ETH. The EIP claims "This ensures that only ETH can ever be used to pay for transactions on Ethereum" I do not see how this excluded extrinsic payment method mechanisms. But without this the burning feature is avoidable by just using some other token for gas.
The proposal splits up the gas being paid into two buckets. One that you, the miner, is paid for processing the transaction and that is variable and controlled by the rate discounting. Another part, the base value, is sent to a dead letter wallet. Thus a small tax is paid to nobody by the miner instead of captured as value from performing the mining.
The burning of the base gas value This is intended to control ETH inflation according to the justification in the EIP. The burning system, presuming someone cannot actually get access to the dead wallet addresses and make themselves instantly rich, creates deflationary pressure.
Inflation is reduction in value over time of the token as a value holder of currency, Inflation is generally used in an economy to encourage people holding currency to spend their money into the economy instead of holding onto it. Basically a "use it or lose it" scenario where money in savings becomes less valuable over time but investments continue to track or beat inflation.
Deflation in any value proxy token, fiat money or bitcoin or anything else with floating value, is very very bad for users. But it is great for passive holders and token hoarders. Collectors do nothing useful and see their coins gain value every day. Those who actually use the token watch as the system burns their tokens. This actual users are penalized. They could do better by just sitting on their coins instead of using them for value gaining transactions.
This could be seen as a rate limiter on use of the network. Or as a 'screw you, I've got mine' proposal. This is the wealthy taxing the poor for the later having to use their money to survive.
Burning also complete irrelevant to the features for controlling the market price of gas through discounting. The burning feature only requires the ETH type to be forced to give it the teeth to be useful. Otherwise nobody would want to use ETH for gas as some of the price tag on the processing is being lost to the miner as a tax.
The former is a reasonable idea. It pushes the gas price toward the low price. That reduces miner profitability but still enables them to big the price up as needed. This can encourage efficient miners and push out inefficient ones, presuming it doesn't result in too many idiots (i.e. those playing the game at a loss.) The later burning feature is just a money grab by the already invested.