- The teacher remarked on how quickly the students were learnig.
- The teacher remarked how quickly the students were learnig.
No1 is an example sentence of the verb "remark".
If no2 is correct in grammar, what's the difference between the two in meaning?
They are not identical in meaning, though in some cases they will be interchangeable.
To remark something is to state it in a remark. If someone "remarked how blue the sky was", they said something like "gosh, the sky is blue".
To remark on something is to make a remark related to it. If someone "remarked on how blue the sky was", they might have said "gosh, the sky is blue". They might also have said something much more elaborate about how the sky being blue reminds them of some anecdote, or why the sky is blue, or whatever.
In your example, a remark on may well have just been a comment that the students were learning quickly, or it could have been a digression of some sort.
To elaborate a little bit on @SamBC's answer:
The teacher remarked how quickly the students were learning.
This is the less ambiguous version. You could replace the word "remarked" with any other speech verb — "said", "exclaimed" — and pretty much preserve the meaning of the sentence. There's a strong implication that the teacher's remark was exactly "How quickly you're learning!" The students are probably learning quite quickly.
The use of the verb "remarked" (as opposed to "said") connotes that the students' rate of learning is moderately surprising to the teacher. Their rate of learning is remarkable (for some reason).
The teacher remarked on how quickly the students were learning.
The teacher went out of her way to make a remark on the students' rate of learning. (In this context, the word "on" means "concerning" or "regarding".) We don't know what the remark was. Maybe it was "How quickly you're learning!", or maybe it was "You're learning at an unfortunately rapid pace and soon I'm going to run out of prepared class materials," or maybe it was "You're not learning as quickly as I'd hoped." Again, though, we can be sure that there is something remarkable concerning the students' rate of learning.
The fact that this is a teacher also raises the possibility of a pun. I don't think it's relevant here, but consider if the two sentences were:
The teacher remarked on the students' papers.
The teacher remarked the students' papers.
Here the first sentence, as usual, means that the teacher made a remark regarding the students' papers. But the second sentence should clearly be understood to mean that the teacher gave new marks to the students' papers: she'd already marked them once, but now she was re-marking them.
The meaning is identical
I don't recall having heard anyone use the second example. However, it's grammatically correct.
The word "on" in your first example is a preposition used as a function word "to indicate the subject of study, discussion, or consideration" (Merriam-Webster, preposition definition 9d). Function words are often used colloquially, meaning native speakers will often use them rather than an alternative without them (for example, your example #2).
Which was a long and complex way of saying your example #1 is how I'd say it.
