"These are my children who want me to..." or "It's my children who want me to..."? And why?
One of the reasons you're unsure of the proper verb is that you've replaced the subject with the pronouns that don't always seem to follow the rules. Both are grammatical.
My children want me to learn to cook.
Want is the 3rd person plural: they want.
In each of your constructions, you're splitting the noun phrase my children into two phrases. You are duplicating my children with two pronouns: These and it. The use of it in this way is called an it-cleft (There is also a wh-cleft).
In the first case, you have represented the subject with a pronoun that is equivalent in number:
These are my children (who want me to learn cooking). (Easier to understand agreement between the pronoun and the verb.)
It is singular, therefore when you emphasize your children with it, you will end up using a singular verb: it is while retaining the proper number in the verb want (they want.) The pronoun here (it) is a dummy pronoun, therefore is not obliged to agree with the noun the it cleft is used to emphasize in number.
It is my children who want me to learn cooking.
vs.
It is my child who wants me to learn cooking.
Also, here is an obvious use of the it-cleft, and the dummy pronoun when emphasizing something which is plural. (Particularly appropriate today.)
Military leadership of the victorious Allied forces in Western Europe during World War II invested Dwight David Eisenhower with an immense popularity, almost amounting to devotion, that twice elected him President of the United States. ...It was the Democrats who gave Eisenhower the votes he required to pass key legislation. - NYT
Richard Norquist explains:
"[A Cleft is] a construction in which some element in a sentence is moved from its normal position into a separate clause to give it greater emphasis... The extra focused element normally appears early in it-clefts and late in wh-clefts."
- I want a cheeseburger. (normal)
- It's a cheeseburger I want. (it-cleft: emphasis early)
- What I want is a cheeseburger. (wh-cleft: emphasis late)