Is it obligatory for the young man to explain to the girl's brothers what happened between them in order to marry her, after her family refused their marriage, and is he entitled to marry her after she lost her virginity because of him?

You must sincerely repent from this sinful and immoral relationship. What happened demonstrates the wisdom of the Shariah in forbidding the taking of secret lovers (akhdan), as mentioned in the Quran. You are not obligated to marry her. If both of you repent and you find a way to marry her, then do so. Otherwise, leave her. You are not permitted to tell anyone what happened, as it is obligatory to conceal your own sins because publicizing a sin is another sin.