Belly putters and long putters themselves remain perfectly legal; the proposed rule change does not affect the length of putters. The proposed new rule, which will be known as Rule 14-1b, is needed, the R&A and USGA say, to maintain the idea essential to golf that a stroke consists of holding a club away from one's body and swinging it freely.
The governing bodies were moved to act now, they said, due to the growing number of golfers (and instructors) who view anchoring as a preferred method of putting, rather than as a last resort; and because they were beginning to see anchoring creep into other areas of the game, such as anchoring hybrids for chip shots around greens.
Belly putters and long putters remain legal to use, even after the proposed rule takes effect, so long as the golfer does not anchor them. So, for example, senior golfers with bad backs can continue to use broomstick putters so long as their top hand on the grip is held away from the body, as opposed to being anchored against the chest or chin.
Here are links to some great explainers of the proposed Rule 14-1b on the USGA website (you can also find them on the R&A website):
Infographic explaining the rule
Video explanation of the rule
Photo gallery of putting strokes that will still be 'legal' under the new rule
Photo gallery of putting strokes that will be prohibited by the new rule