It is actually quite simple: if you hold different tokens across chains and want to perform a swap or a bridge, you need to have the gas fee token of the source chain.
For example:
You have USDC on the Avalanche chain and want to use it to buy a meme token on the Robinhood network. What you need is the AVAX token; that is what will execute the transaction. This means the gas fee token.
You have completed the transaction and now hold a meme token on the Robinhood chain. Now, you want to sell that meme token:
Since the token is now on the Robinhood network, you will need the gas fee token for that network , which is ETH.
Your tokens are the assets you hold; your gas fee tokens are like the processors.