The idea is that Bitcoin is agnostic when it comes to any sidechains derived from it. When some coins are brought in from Bitcoin to the sidechain, Bitcoin only sees what it always sees: transaction outputs locked up by an encumbrance script. The other chain must be aware of Bitcoin's blockchain, and from there, the sidechain's rules will see that the coins have been encumbered such that the coins are now issued on the sidechain's blockchain.
So let's say that Ethereum does not issue ether via mining reward, but rather requires you to lock up bitcoin on Bitcoin's blockchain, thus "moving" that bitcoin from the main chain to the sidechain. Now the ether can be traded, payed to a smart contract, or anything else that ether can do on the Ethereum block chain. You can even trade it back to the main chain, thus claiming the bitcoin that was "moved" to the Ethereum chain. The fact that the two blockchains are fundamentally different doesn't matter, as long as the sidechain is aware of the parent, and can handle moving coins back and forth.
You may want to check out this answer for a more in-depth explanation to sidechains.