Learn Italian: Coloro che ascoltano un risparmiatore sanno gestire i soldi. - Those who listen to a saver know how to manage money.

Yes, absolutely. Quelli che and coloro che both mean 'those who'. The difference is entirely about register. Coloro che is more formal and elegant, often found in written text, speeches, or proverbs. In everyday spoken Italian, you are much more likely to hear quelli che ascoltano....

Why is 'know how to manage' translated as just sanno gestire without the word for 'how' (come)?

In Italian, when you want to say you have the acquired skill or knowledge to do something, you simply use sapere followed directly by an infinitive verb. Adding come (sanno come gestire) sounds like a direct English translation and is generally unnatural; sapere already includes the 'how to' meaning in this structure.

Why is there no preposition after ascoltano? Shouldn't it translate to 'listen to'?

Unlike the English verb 'to listen', the Italian verb ascoltare is transitive. This means it takes a direct object without needing a preposition. You just say ascoltare qualcuno (to listen to someone), so here it connects directly to un risparmiatore.

How is the noun risparmiatore formed, and what is its feminine equivalent?