How do other nodes know my balance with LN?
The two channel owners have cooperatively locked up a sum of Bitcoin in a 2-of-2 multisig address.

The only thing that they can do, is to renegotiate how much each of their balance shares is. This is what happens when they send funds in Lightning: the sender offers to increase the balance share of the recipient and at the cost of reducing their own.

Likewise, when you attempt a multi-hop payment, you offer a conditional smart contract to your channel partner. Your channel partner gets to increase their channel balance share at your cost, when they prove to you that they have forwarded the sent sum to the recipient.

Since the sum of the two balance shares can never exceed the channel capacity, nobody can ever send more money than their balance share: Their channel partner would simply deny the payment attempt.