How can an atheist explain to a religious person that there is no god?
You could say what you just asked. Although I'm more likely to pick a fight with you for making a claim you can't back up.
I might remind you that the onus is not on you to take that position. They're the ones asserting something that flies in the face of most of observable, demonstrable reality. If they can't successfully make their case, that's not our problem.
But if we do push further - how can you convince a theist that their position is wrong? My position is that this isn't the right approach. There's two things a person needs to successfully deconvert:
- A solid epistemological toolset - so they can accurately assess reality. Being aware of common logical fallacies can help, like some sort of quick spelling/grammar check on your own thinking.
- They need to care.
You can help educate a person, piece by piece, on #1, but unless the person cares that what they believe is actually true, and/or that their fundamental approach to knowing reality actually works... you're going to faceplant into a concrete wall. There's no magic sequence of words that'll work. Most atheists I'm aware of, who were former theists, cared. They dug into what was told to them their entire lives, and found it wanting, and stopped being convinced by it.
For many, they won't give up their security blanket for the world. Sometimes it's more practical. The secular part of society doesn't hold a candle up to the long-time organized social infrastructure of the religious... even if we're talking about something as straight forward as daycare. That's improving, though. But it's going to be difficult to argue someone out of giving up their free daycare, because technically we're right.
Ultimately, if the person does start caring, they'll do your work for you. Maybe after all these years, I'm just this cynical about the capacity of humans to be rational, but it's most likely going to be the case. It all comes down to values, and whether they've landed in the right place in a person's life.
[I think this could be a decent front-page question, so I'll recycle my response there]