Lightning: Why would Bitcoin Network Accept a Breach Remedy? The Lightning Network Whitepaper discusses a situation in which one party of a payment channel broadcasts an outdated commitment transaction.  The other party through a breach remedy transaction will get the cheating party's coins.

How does this work though?  If the dishonest party broadcasts a transaction to the Bitcoin network, isn't that transaction more likely to be accepted (since it was first) than a subsequent (up to date and legit) transaction?
You are right. If the dishonest party publishes an outdated commitment transaction it is most likely be accepted by the Bitcoin network. 

The key trick is that the old state has an output script that gives the other party the opportunity to spend all outputs (even the one that was supposed to belong to the dishonest party) with the so called breach remedy transaction. This mechanism works via a time lock in one output of the commitment transaction and is called a Revocable sequence maturity contract (RSMC) 

So if Alice and Bob have a channel with 5 mBTC for Alice and 7 mBTC for Bob. But Alice decides to publish an old commitment transaction in which Alice has 9 mBTC the outputs of that tx look like this :


Bob can immediately spend 3 mBTC
anyone with a Revocarion secret can immediately spend 9 mBTC 
alice can spend 9mBTC after a timelock of for example 144 blocks. 


The breach remedy transaction is the transaction spending the revocable output. So it is not an alternative to the fraudulent commitment transaction but rather a guaranteed way to spend the output of the commitment transaction that is supposed to belong to the dishonest person. It is guaranteed as the timelock prevents the dishonest person from spending their output.