Q: Does "I owe explanation to no one = I don't owe anyone any explanation" same? I get confused about using "not/don't with "no one/anyone/anybody". Like I want to mean "I don't owe anyone any explanation" in short. Hence I used "I owe explanation to no one". I'm still confused if it's correct?

A: 
I do not owe anyone an explanation

means exactly the same thing as

I owe no one an explanation

Thus, I am not even sure what you are asking
not anyone = no one
Your sentence

I owe explanation to no one

is incorrect, but not because of "no one." It requires an indefinite article before "explanation. What is correct is

I owe an explanation to no one.


A: First let us correct this sentence.

I owe explanation to no one. (incorrect)
to
I owe an explanation to no one (correct)

The first sentence was missing an article. Articles are "a", "an" and "the".
Next, generally we can switch the phrases around. Thus

I owe an explanation to no one = I owe no one an explanation
I don't owe any explanation to anyone = I don't owe anyone any
explanation
I don't owe any explanation to anyone = I do owe an explanation not to
anyone

Notice the difference? "do" is used for emphasis. And "any" works nicely with "don't" for the same purpose, emphasis. Otherwise they are the same. Thus

I owe an explanation to no one = I owe no one an explanation
are almost equal to
I don't owe any explanation to anyone = I don't owe anyone any
explanation

