There two different places where the '-ing' form of a verb is used.
One is called the *gerund", which is the noun form of a verb.
Let's dissect this sentence from your example:
Education minister, Adam Miller, explained that the government would make a number of radical changes such as introducing smaller class sizes and providing more funding for computers and books.
The subject is "the government" and the verb is "make". "Introducing smaller class sizes" and "providing more funding" are nouns. (Techncially, noun phrases).
If it's not clear why these are nouns, let's simplify the sentence:
The government would make X and Y.
This is a simplified version of the sentence, which hopefully makes it clear that X and Y are things the government makes.
Given that "introducing smaller class sizes" functions as a noun, we need to use the noun form of the verb "introduce", which is the gerund "introducing".
However, Mr.Miller put off bringing them in until now because of concerns with some of the details of the proposal.
"Bringing them in" is a noun phrase here, as well. The verb is "put off". So again, the gerund form of "bring" is appropriate.
The OP's example also demonstrates the second use of "-ing", which is the continuous tense. The continuous tense refers to an action that is, was, or will be ongoing.
This sentence is using the future continuous tense:
He said the new system should be operating by June at the latest.
If "the new system" is a computer system for school administrators, then such a system typically runs without interruption for years, so it makes sense to use the continuous tense to describe that.
(N.B. There are a number of technicalities related to the continuous tense that I won't go into here.)