How does muun lightning wallet requires no trust and no backups? https://muun.com/ claims it has a wallet that does not rely on backups.
Here: https://blog.muun.com/why-not-just-a-mnemonic/ it says

Muun is a multisig wallet with lightning support, so backing up a
mnemonic alone would mean your bitcoin could be easily frozen.
Self-custodianship is at the core of everything we do, so this wasn't
acceptable. We built an Emergency Kit that gives you full self-custody
of your money when combined with your Recovery Code[6].
Your Emergency Kit is a PDF document with the information and
instructions needed to find and spend your funds independently. No
need to brute-force any missing data. Everything is there, including
your private keys and output descriptors[7].
Your private keys are securely encrypted with your Recovery Code. This
makes your Emergency Kit harmless by itself: you can keep multiple
copies, and you can even store it safely on the cloud.

How can it, however, keep track of the state of my lightning wallet? If I use it to only spend, then there's no problem. However, if I receive more than I spend, it's of my interest to save the transaction state so I can propagate it when I want to close the channel.
I don't see how this emergency recovery kit can help me recovering the state.
I'm the CEO at Muun wallet.
Muun does rely on backups since the Emergency Kit is a backup, just not using mnemonics. The problem with mnemonics is that they don't contain a lot of stuff needed for the recovery of a multisig / lightning wallet, like the derivation paths, scripts, etc.
When using Muun, you will back up two pieces of data: the Recovery Code and the Emergency Kit. The Recovery Code is a randomly generated code that you write on a piece of paper, in the same way that you usually do with mnemonics. In fact, this code has higher entropy than a 12-word mnemonic. The Emergency Kit is a PDF document with all the stuff needed for recovery but harmless by itself. It has:

The two private keys (Muun is a 2-of-2 multisig), encrypted using the recovery code.
The output descriptors, with the derivation paths and scripts.
Instructions on how to use this data to recover your funds.
All this data is also available inside the PDF in a machine-readable format.

When using the recovery tool (https://github.com/muun/recovery), you will be able to move the funds using your Recovery Code and Emergency Kit combo. The next version of the recovery tool will use the machine-readable metadata in the PDF for easier input and better UX.
Regarding the lightning aspect of the recovery process, Muun uses super optimized submarine swaps to execute the lightning payments, both incoming and outgoing. This means that your funds are on-chain and thus recoverable by the mechanism I just described.
From the next version, Muun will start maintaining off-chain funds, and thus it will need to backup the off-chain commitment transactions. These transactions will be included in the PDF metadata. I'll update here with more information once we release this.