What is the legal ruling on someone who broke their fast for a number of days in Ramadan due to the excuse of travel, then became negligent in making them up until the next Ramadan began? And how should they make up those days? And is it sufficient for them to pay the monetary value of feeding the poor if they are required to feed the poor in addition to fasting?

Your friend must make up all the days he missed. If he delays making them up without a valid excuse until another Ramadan begins, then in addition to making them up, he must feed a poor person for each day. It is permissible to give the monetary value instead of food for expiation if this benefits the poor person, as some jurists have allowed. Shaykh al-Islam Ibn Taymiyyah holds that giving the monetary value without a need or a preponderant benefit is forbidden, but giving it for a need or benefit is permissible.