I had a huge explanation typed up, and then lost it. Damn it.
Essentially, what this proposal would do is eliminate the domestic short season leagues: the Appalachian, Pioneer, Northwest, and New York-Penn Leagues. Officially, the Appy and Pioneer Leagues are Rookie level, while the Northwest and NY-Penn are Class-A Short Season, but in practice there's no difference. These leagues start their seasons in mid-June so that newly drafted rookies and second year players who weren't ready for a full season (and got stuck in extended spring training) can start playing meaningful games right away. Unlike full-season leagues, there is no one-and-only-one requirement here, and subsequently some teams have two affiliates (though none have two in the same league), resulting in 40 teams across the 4 leagues. (For reference, there are 120 full season affiliates, 30 at each of AAA, AA, High-A, and Low-A. That's where the "160 minor league teams" number seen in stories comes from, the sum of short season teams and full season teams. )
Draftees that aren't ready for competitive ball go to the parent club's spring training complex and play in what are essentially summer instructional leagues. While those "complex leagues" do keep records, they don't charge admission, are usually played on side fields, and are only seen by scouts and family members. Some teams also have multiple squads if they have enough players, meaning that there's some 40 teams at this level – the number actually varies year-to-year, as teams decide whether to field one squad or two somewhat ad hoc. (Add in the academy teams in the Dominican Republic for young guys from Latin America, which every club has at least one and many have multiple, and the total number of affiliated teams is actually around 240, though only the 160 teams that charge admission really play competitive baseball.)
The proposal would move the draft to August from early June, and cut the length in half. Without the time or the players to staff them, the short season leagues would go away. That they're usually played in tiny, often rundown ballparks in small towns is part of the reasons given as well. Draftees would essentially do little more that scouting exercises for a few weeks and not playbill actual competitive games until the following season.
Beyond that, there's also some reorganization proposed to reduce travel, reduce distances between MLB clubs and their affiliates, and do away with some of the poorest ballparks at higher levels by replacing them with teams from the short season leagues that do have good ballparks. Strangely enough, the plan calls for taking over two independent league teams (St Paul Saints and Sugar Land Skeeters), despite the fact that they have zero power over them.
It will be interesting to see how much of the proposal survives intact. Supposedly the MLB side had a very "take it or leave it" attitude in the initial discussions. But the backlash has been swift. There was letter signed by over a hundred members of congress criticizing the plan. No one really wants to lose their local team, and 42 teams is a lot to drop.
That said, I can see the independent leagues jumping into some of the abandoned markets. Despite the erroneous info above, there is nowhere near that many independent league teams. There's actually only 62 in 7 leagues, and the quality and format varies hugely depending on the league, with an informal hierarchy that's widely recognized among players and scouts. The nicest independent league ballparks are better than many of the affiliated ones, and the quality of play in the top leagues approaches AAA levels. One can see why affiliated ball might want to poach some of those.