This law doesn't primarily address the awarding of H1B visas It's about "immigrant" visas.
There is some tightening of H1B management in this bill (slipped in by Sen. Grassley), but mostly it isn't designed to address non-immigrant immigration to the US at all ( e.g. H1B, L1, O1 non-immigrant employment petitions). It addresses the per-country visa caps for "Immigrant" visas - e.g. applications for permanent residency in the US ( commonly known as green cards ). For most primary applicants, this involves a labor market test ( is there a US Citizen / Green Card holder who has the minimum skills to do this job) supervised by the department of labor.
This bill addresses per-country limits to the number of visas issued each year which had caused countries with large populations and a tradition of emigration to the US to have decade-long backlogs in the time to adjudicate and issue a visa. Most of these people already work in the US under non-immigrant visas ( H1-Bs can be extended each year if there is a pending immigrant application). It probably does more harm to US workers salaries to keep these non-immigrants in 'zombie' positions within the company sponsoring their non-immigrant visa, rather than have them boost the average worker's salary through open-market employment opportunities.
The controversy about this bill has been how to ensure immigration from the rest of the world (RoW) is still possible while they run down the backlog of green-card applications for applicants born in India, China, Mexico and the Philippines. The Senate seems to have found a compromise they are happy with. But it must be reconciled with the house bill ( HR 1044 ) and signed by the president to become law.