I know there are many sentences like I went to sleep at six and I slept at six, but I want to know why it is incorrect to say "I slept at six" in English.
In my English, at least, sleep can be specified with a period or length of time, but not a point in time. (If you supply a point in time, it's intelligible, but doesn't sound natural.)
So I could say things like
I slept from nine until seven.
I slept until noon.
I slept for thirteen hours.
But if I want to tell you what I was doing at a precise time, I need to say something like
I was asleep when the fire alarm sounded.
Or I could tell you about the point when I went from waking to sleeping:
I went to sleep around midnight.
I fell asleep as soon as my head hit the pillow.
It's not incorrect. It's a perfectly valid sentence construction which is commonly used.
I slept at six.
I got up at six.
I ate at six.
In general, the other answers do a fair job of explaining why it's at least weird to use "I slept at [time]" to describe the time you began sleeping. But there are plenty of occasions when it works well enough.
A short nap could be described as "I slept at noon." This works especially when you're describing a sequence of events, e.g. "I slept at noon, then had lunch at 1."
Similarly, I wouldn't be confused if someone summed up their evening like this: "I jogged at 6, ate dinner at 7, and slept at 9." (i.e. they went to bed at 9)
If I asked someone "When did you go to bed?" I'd understand what they meant if their response was "I slept at 10."
However (thanks Dog Lover [Love the handle, BTW]), the way that sentence is constructed is a bit unexpected. Consider, "I went to bed at six".
"I slept" means "I spent a period of time asleep." "I slept at 6" doesn't make sense because it's saying "I spent a period of time asleep at a particular moment in time."
In contrast, "I woke" means "I changed from the sleeping state to the waking state" and it's perfectly reasonable to do that at a specific time.
Having said that, "I ate at six" is fine, even though it seems to mean "I spent a period of time eating at a specific time." It would, instead, be interpreted as "I began eating at six", which is again a change of state at a specific time. This is somewhat inconsistent but that's how language is: all I can offer is that "to sleep" isn't used to mean "to begin sleeping" even though it would be logical if it was.
"I slept at six" is not incorrect. It is a bit unusual, and feels a bit clumsy, and it means something different than "I went to sleep at six". That happens quite often, some sentence is not incorrect as a sentence, but has a different meaning than you think it has.
"I went to sleep at six": I was awake at 5:59, but asleep at 6:00.
"I slept at six": I went to sleep some time before 6:00 and woke up some time after 6:00, so at the exact time I was asleep.
Example usage: "I called you on the phone but you didn't answer". "When did you call?" "I called at six. What were you doing that kept you from answering?" "I slept at six, that's why I didn't answer".
Usually you would say "I was asleep" instead of "I slept".
It's a little ambiguous as it's unclear whether you mean you started sleeping at six, or slept through six, though it would probably become clear from the context or question.
I world use "I went to sleep at 6" or "I was still asleep at 6" to make it clearer.