This opinion, which is that of Imam Shafi’i and Imam Ahmad, uses as evidence the same verse as the other scholars who agree that these women need to make up the missed days:
“So whoever sights [the new moon of] the month, let him fast it; and whoever is ill or on a journey – then an equal number of other days. Allah intends for you ease and does not intend for you hardship and [wants] for you to complete the period and to glorify Allah for that [to] which He has guided you; and perhaps you will be grateful.” [Al-Baqara, 2:185]
As for the paying of the fidya, Imam Shafi’i and Imam Ahmad use the actions and rulings of the companions regarding this issue; the companions would not have ruled in favor of the fidya had they not heard the Prophet (pbuh) himself say that it needs to be paid.
The majority of the scholars do not follow this opinion that a fidya needs to be paid because there is no clear evidence of the Prophet (pbuh) affirming this.