The first sentence of the studentaid.gov page has a link to a Congressional Research Service report on the statute. The CRS is a congressional agency (part of the Library of Congress) which writes many reports of this kind for the aid of legislators and the public.
The second paragraph of that report says:
The FAFSA Simplification Act (FSA; Title VII, Division FF of P.L. 116-260) makes significant
changes to the underlying processes and methodologies for determining federal student aid
eligibility. As originally enacted, the FSA had a general effective date of July 1, 2023. In March 2022, the FAFSA
Simplification Act Technical Corrections Act (FSATCA) was enacted as Division R of the Consolidated Appropriations Act,
2022 (P.L. 117-103). The law pushed the general effective date for the FSA back one year, to July 1, 2024, coinciding with
the beginning of the 2024-2025 award year. Both the original FSA and the FSATCA authorized ED to implement certain
provisions prior to the general effective date.
The citation "P.L. 116-260" is to the Consolidated Appropriations Act, 2021 (link is to a 2126 page PDF representing about half of the act), and contains the FAFSA provisions at the location stated. This differs from S.2667 because that was a version of the bill as introduced to the Senate, and due to the political process quite a lot of legislation gets shunted into these colossal budgetary laws. I haven't compared them line-by-line so I don't know if there are any differences between the versions.
Within the text, Section 702 of the FSA enacts various amendments to the Higher Education Act, 1965. There are different applicable rules depending on whether the student is a dependent, is married, has dependents of their own, etc.; I'll look at just the dependent case since the other ones are parallel.
Section 475(b) of the Higher Education Act, as amended and codified at 20 U.S.C. 1087oo, computes a quantity called "parents' contribution from adjusted available income". This is used to figure out how much of the student's costs could be met by the parents. Among the several steps of the calculation, there are two that take account of other family members being in college. One is an "income protection allowance", a deduction from the parents' gross income. This varies depending on the size of the family and how many of those people are in college. And the final step is to divide the amount computed so far by the number of family members in college.
In the replacement version, the income protection allowance is based only on the size of the family (not on how many of them are in college), and the final quantity is left alone rather than being split among the college-goers. This is all specified in 702(d) as a drop-in replacement for Section 475 of the HEA (and again there are parallel rulesets for students in different situations), on p1962 of the PDF linked above.
On the face of it, the new calculation may seem a bit odd since it spits out an amount of money that the parents notionally have available, but then assumes it can be spent on all of their college-going children at the same time. However, the CRS report explains (see sections "Income Protection Allowance" and "Families with More Than One Member in College", pp6-8 and footnotes) that because these allowances are more generous across the board, it should still work out in favor of the students/families.