EIP-1559 introduces a number of things and there's many explanations, and here's one approach at explaining.
EIP-1559 allows simpler UX for fees and improves the consistency of transactions being mined in a block. EIP-1559 also reduces overpayments to miners.
First, EIP-1559 introduces a flexible block size, and it targets usage of 50%. This means most of the time blocks will be half-full. There can still be spikes when there will be full blocks for some period of time, but that's only likely for short time periods.
Since blocks are usually half full, this means that for most users, all you need to specify is a very small tip (aka priority fee, possibly 1 gwei per gas), and your transaction will still get mined within the next few blocks. Unless you are sending a transaction when the blocks are full, everyone using the minimal tip, will work out well for most people. (Of course if everyone tries to outsmart each other and use 2 gwei, then everyone has to use 2 gwei: so I suggest not increasing your tip unless you really find it's too low.)
To give you a concrete example, you may have seen gas prices recently fluctuate from 5 gwei to 10gwei and higher. Right now you have have to guess what number you want to use, 5, 10, or higher. With EIP-1559, you just set your tip to 1 and your transaction will get mined very soon in the next handful blocks.
The other problem without EIP-1559 is you might see a gas price website which tells you 5 gwei. So you transact with 5 gwei and wait minutes and minutes because the gas price went up to 7 gwei, usually because some people didn't know better and transacted with 7 gwei. So your transaction waits to get mined until 5 gwei is accepted by a miner. In EIP-1559, if everyone tips 1 gwei, everyone will still usually get in very soon because remember that blocks are flexible and can double in size.
How does EIP-1559 reduce overpayments?
Assume a base fee of 100 (gwei).
EIP-1559 introduces a new transaction type. This allows you to do something like specify a tip of 1 and a
max_fee_per_gas of 150, and still pay only 113.5, if the base fee in the next block jumps to 112.5. (A full block increases the base fee by 12.5%.)
Without using the new EIP-1559 transaction type, you would pay 150.
You might set a
max_fee_per_gas of 150 because blocks are full and you are willing to pay up to 150 for the chance to have your transaction mined. This is very different from specifying a tip of 50.
Why is the base fee burned?
The EIP offers a solid explanation:
This ensures that only ETH can ever be used to pay for transactions on
Ethereum, cementing the economic value of ETH within the Ethereum
platform and reducing risks associated with miner extractable value
(MEV). Additionally, this burn counterbalances Ethereum inflation
while still giving the block reward and priority fee to miners.
Finally, ensuring the miner of a block does not receive the base fee
is important because it removes miner incentive to manipulate the fee
in order to extract more fees from users.
A little more information about why base fee is burned
EIP-1559 was proposed because first price auctions are inefficient. The base fee needs to be burned to prevent off-chain markets by miners that would circumvent the base fee and degenerate to a first price auction.
https://timroughgarden.org/papers/eip1559.pdf briefly explains 7th takeaway:
The seemingly orthogonal goals of easy fee estimation and fee burning
are inextricably linked through the threat of off-chain agreements.
In https://youtu.be/62UI3Js30Io?t=1935 Roughgarden gives an example of how a base fee of 90 does not prevent transactions that pay less than 90. Because a user and miner can have an off-chain agreement where a user who only wants to pay 60, on-chain they pay 90, but get refunded 30 off-chain by the miner.
Some other references:
https://www.youtube.com/watch?v=IEQPSJbZsq0 - EIP 1559 Overview
Note: because of MEV the tip probably needs to be more than 1 gwei:
2 gwei is probably a very good default “you will get in the next handful of blocks” value. 1 gwei is likely a good “slow” value.