[annalex]

How does Catholic or Orthodox Church explain Roman 6:10 [...] It asserts that Christians don't sin

I don't think it asserts that at all. It says that Christ dies for our sin and that we should become dead to sin.

[Kolokotronis]

"How does Catholic or Orthodox Church explain Roman 6:10 [...] It asserts that Christians don't sin.

I don't think it asserts that at all. It says that Christ dies for our sin and that we should become dead to sin."

Like I said earlier, read +John Chrysostomos' Homilies X & Xi on Romans. Seems pretty clear to me.