Selecting some of the OED's definitions in chronological order, I see a three step process (my numbering):
To give real existence to something.
To present as real to the mind; to make to seem real; to bring vividly or clearly to mind as if real
To conceive of as being real; to apprehend with the clearness or detail of reality; (in later use also) to become aware of or come to understand (a fact, situation, etc.).
In other words, the original meaning is to intentionally bring into reality;
the second is to intentionally imagine that this were so; and, finally, to come to mind unintentionally
I realized (made real) A.
I (intentionally) realized (made real in my mind) A. = to "build" a picture.
I (unintentionally) realized/came to realize (that) A.
In the end, A is no longer one thing, but an idea/concept requiring a clause. A clause is required because what we are realizing becomes more complex, for example an entire situation, as the OED definition states in (3) above, which sums up this transformation in meaning. Meaning (2) is the fulcrum, making the process a mental one.
The that clause also nicely serves to distinguish (a) from (b) and (c):
(a) I realized my dream
(b) I woke up and realized it was a dream
(c) I realized it was a pipe dream.
It would be interesting to see whether we really can sharply divide all usages into (b) suddenly realize and (c) come to realize. We can usually tell from the context/timespan, but I think there are instances that blur the line, as, perhaps, in the following examples I've found.
It was easy to pick on poor, weak homeless people. I was trying to be
as normal as possible, but for the police I was homeless. That's when
I first realized I was one of “them” in their eyes. S. C.
Greenfield; Sacred Shelter: 13 Journeys of Homelessness and
I don't remember exactly how or when I first realized I could sing
like Elvis—maybe in the shower, maybe in the car, I don't know...
Matthew Vollbrecht; Blind Faith
I think the adjective first in these examples refers to the beginning of a gradual coming to realize process but also, perhaps, the first a series of discrete "realization events," which may each be more or less sudden, but, one would think, gradually less surprising.
"Where are we?" he asked.
The man didn't respond. Dawkins blinked,
looked left and right, and slowly realized he was in a small jet
that was airborne and probably had been for a while. D. Mann and R. Pezzulo; SEAL
Team Six: Hunt the Dragon
Here IMO the slowly puts us in a no-man's-land (no-person's-land ?) between the eventive and stative senses as E.A. refers to them in his comment below. Would there be much difference if the time period were shorter by seconds or minutes (...and suddenly realized...), aside from clearly pushing the meaning to the eventive?