Look at the following two sentences.
Example sentence 1] he improved their house(He improved their house.) Collins Dictionary of Senior Scholars].
Example sentence 2] we need to improve on our performance against France(We need to improve our performance against France.) Collins Dictionary of Senior Scholars].
There is no problem with [Example Sentence 1], but there are three problems with [Example Sentence 2]:
1) Can on be removed?
2) Is there a difference between improve on and improve?
3) Is improve in [example sentence 2] an intransitive verb?
Question (3) is relatively easy because the intransitive verb is the subject, such as.
Example sentence 3】The overall situation has improved dramatically(Overall, the situation has improved significantly.) [Oxford Dictionary of Senior Scholars].
Example sentence 4】things are improving every day(The situation is getting better every day.) [Oxford Dictionary of Senior Scholars].
Example sentence 5] working conditions h**e greatly improved(Working conditions have improved considerably.) [Oxford Dictionary of Senior Scholars].
Example sentence 6】His quality of life has improved dramatically since the operation(After the surgery, his quality of life improved significantly.) [Oxford Dictionary of Senior Scholars].
Example sentences 3-6] are improved, but [example sentence 2] is definitely not the subject we is improved, but our performance. So it can't simply be said that improve is an intransitive verb.
Because our performance in [Example sentence 2] is improved, the syntax is fine after removing on, and the semantics seem to be similar to on.
Let's take a look at the following sentence.
Example sentence 7] we've certainly improved on last year's figures.(We're definitely an improvement over last year's numbers.) [Oxford Dictionary of Senior Scholars].
Obviously, the on of [example sentence 7] cannot be removed, because there is a problem with the semantics after removal, because "last year's number" is a fait accompli, how can it be improved!?
In this way, if on is followed by "fait accompli", it cannot be removed.
The following are example sentences in the mainstream dictionary, example sentence 8] the company is hoping to improve on last year's sales figures.[Cambridge Academic Content Dictionary].
Example sentence 9】Last time she ran the race in 20 minutes, so she's hoping to improve on that.[Cambridge Business English Dictionary].
Example sentence 10] bertorelli has scored 165 points, and i don't think anyone will improve on that.[Longman Dictionary of Contemporary English].
Example sentence 11】The government hopes to improve on the 3% economic growth of the past two years[Longman Dictionary of Contemporary English].
Obviously, the above example sentences are all "existing facts" after on, and on cannot be semantically removed.
Once you have grasped the above essence, you will be able to understand the definition of improve in the Longman Dictionary of Contemporary English - to do something better than before or make something better than before.
Obviously, improve is followed by the object noun that is improved, and improve on is followed by the object's immediate information.
Going back to [Example Sentence 2], with on emphasizes the improvement of performance compared to the past, and without on emphasizes the improvement of performance.
The preposition on may have "in." on the foundation". However, if it is so strongly understood, improve should be an intransitive verb, but there is a conflict between the subjects of the intransitive verb list.
Perhaps improve on is an abbreviation for improve oneself on (oneself is the self of the action subject).
Because of the above conflict, the dictionary simply labels improve on as idiomatic. Since it's a habitual usage, it's natural that you like it, don't analyze it.