The employment rate has continued to rise in big cities thanks to the efforts of the local governments to increase it.
Your confusion is probably increased by the three different uses of the word "to" in that sentence.
The middle one is part of the idiom "X is happening, thanks to Y." This is a relatively informal way of abbreviating the idea that Y is responsible for X. That is, if you want to thank somebody for X (i.e. "give thanks to somebody for X"), you should send your thanks to Y. In this case, the word "to" is being used in its normal prepositional sense ("directed towards").
So we can rewrite the sentence like this:
The employment rate has continued to rise in big cities.
Why? Because of the efforts of the local governments to increase it.
(where the antecedent of "it" is "the employment rate").
You later commented:
I just realize the role is "efforts","of the local governments" is additional burden of this sentence!
Right.
The employment rate has continued to rise in big cities.
Why? Because of efforts to increase it.
Whose efforts? Local governments' efforts. The efforts of local governments.
Incidentally, I don't think "efforts of local governments" is particularly idiomatic. I would rather say
The employment rate has continued to rise in big cities.
Why? Because of efforts to increase it.
Efforts by whom? Efforts by local governments.
The employment rate has continued to rise in big cities thanks to the efforts by local governments to increase it.
We still need to explain the first and third uses of "to" — "continued to rise" and "efforts to increase it." In both cases, as Zebrafish said, "to X" is the to-infinitive form of a verb; that's just how infinitives work in English.
Frequently, you can use either a to-infinitive or a progressive after another verb; so,
The employment rate has continued to rise (acceptable)
The employment rate has continued rising (acceptable)
This is because the thing following "continued" is something like a noun phrase: it says what is being continued. We can use an infinitive, a gerund like "rising", or even an actual noun:
The employment rate has continued its upward trajectory (acceptable)
However, I can't think of any cases where you could do that after a noun (such as "efforts"). This is because the thing following "efforts" is something like an adjectival or prepositional phrase: it says what kind of efforts, or efforts with what purpose.
efforts to increase the employment rate (acceptable)
efforts with an eye to increasing the employment rate (acceptable)
efforts at increasing the employment rate (acceptable IMHO, but informal)
*efforts increasing the employment rate (unacceptable)
And in neither case is it acceptable to omit the "to" part of the infinitive:
*The employment rate has continued rise (unacceptable)
*efforts increase the employment rate (unacceptable)