Turing completeness in layman's terms means you can write programs on it.
Traditionally a computer that most people are familiar with is a single machine, like PC or the device we can hold in our hands. Etherium is a swarm of millions of computers that anyone can add to the swarm.
When people claim Ethereum is turing complete, it means that anyone can launch a program into it, and the "swarm" runs the program. But which node's output is the truth? Without some mechanism in place, a bad actor could claim to have computed an output that serves them greedily.
Ethereum has baked in protocols so that the distributed result of millions of nodes running a program reaches a stasticial consensus.
Being turing complete means that any computable program can run on Ethereum.
So you can imagine writing a program that, say, runs an auction, or performs escrow for contractors, or whatever. You could run doom on it, albeit at an extremely low framerate because you need the swarm to validate each frame.
I hope this explanation helps.
Edit: I swear autocomplete is becoming more forceful over time.