When you answer the phone, I'm wondering if 'This is speaking' is correct instead of 'This is he speaking.' please tell me the correct expression.
You wouldn't normally refer to yourself as "this" without using a personal pronoun in the predicate. I usually say, "This is she," but my usage is probably a bit overly correct. Probably, most people say "That's me." (Even though the objective "me" is technically incorrect here.)
Regardless, "this is speaking" is neither correct nor natural grammar.
Like @joiedevivre, I was taught from an early age to say This is he, but this is rather formal, deriving from centuries of directive that only nominative pronouns should follow a copula. In practice, this is unnatural for a native speaker; a well-educated person who is careful to state that is he in writing will say that's him in casual conversation and not think twice about it. The nominative forms are nowadays used almost exclusively as subjects.
On the other hand, This is he is far better than It is I, which may provoke uncontrollable laughter unless you are quoting the Bible, or perhaps a comic book superhero.
There is no single formula for responding to a question like May I speak to Jinsook? or Is Jinsook available? Any of the following might be acceptable:
This is Jinsook speaking.
You're speaking to Jinsook / You're speaking to her.
This is Jinsook.
This is she.
Jinsook here. (not Jinsook is here)
I'm Jinsook / I am Jinsook.
That would be me.
You can insert Yes in front of any of these, with a pause, and some people will also use them in combination with one another, separated by a pause. Thus, if you have heard other people saying This is speaking, it may have been a conflation of two separate responses that should have been separated:
This is, speaking
This is speaking in any way sounds incorrect to me.
If you are answering the call, you don't refer yourself as a 'third person!' So, it should be 'This is Jinsook Lee speaking.'
However, I'd refrain myself using that, and rather would go with the commonest one:
This is Jinsook...
I definitely concur with Bob, and disagree with Maulik V about not using the third person. On the phone is about the only place I find using the third person about me as a natural thing, normally when specifying who in the household or business is "on the phone".
"ACME Industries, Ted speaking." would be a perfectly acceptable opening, as would:
"ACME Industries, this is Ted"
You could stretch it out by saying:
"ACME Industries, this is Ted speaking"
but you cannot abbreviate it, by removing the name. "This is he" works as a confirmation of who is on the line, but it is still a bit awkward, but "This is him" is even moreso. Talking on the phone does not always involve natural language. We often use hand gestures while on the phone. While that may be natural, it is not very useful.
I'm going to go out on a limb here.
Q: Is this John Smith?
A: This is, speaking ...
makes perfect sense, but certainly isn't all that common. Simply saying "Speaking." as suggested by @CinCout is what I grew up hearing my father say on the phone, though.
I am a native speaker, and I usually say "This is Bob"...I have heard "speaking" from time to time, and also "this is" but I have never ever heard "this is speaking" the construction does not make any sense without the comma as suggested above, but even that would be extremely rare. In informal situations "it's me" would probably be used, especially with younger people, but in business it would be considered very informal
"This is speaking" sounds completely incorrect to me. I would wonder if I had misheard or missed part of the conversation due to a faulty phone line. "This is ..." would normally be used like:
This is fun.
This is a knife!
Personally I answer the phone with my full name and nothing else, so that the caller knows immediately they have reached me (assuming that is their goal). In your case you could say:
Hello, Jinsook here.
Jinsook Lee speaking.
Once again, don't say "This is speaking". If I heard that I might think you had said "Therese speaking" and wonder if I had dialed the wrong number.