I haven't read the paper yet, but from their description, this is apparently based on the previous Rowhammer vulnerability. Rowhammer used a physical flaw in the way high-density DRAM works to allow someone to flip bits in RAM other than those that their program could access directly. As individual DRAM cells have gotten smaller, the chance that they can electromagnetically interfere with each other has gone up. Rowhammer allows an attacker to write to memory they have access to, and from this deduce the contents of other memory by looking at these interference patterns.
RAMBleed is a method to do similar things but without the need to write to RAM at all. This means that ECC (error correcting) memory used in servers does not mitigate the exploit. More recent DRAM4 with a safeguard called targeted row refresh can make the exploit more difficult, but doesn't prevent it.
This seems like a pretty major issue, and one that will be difficult to mitigate. It's unclear what if any software fixes would be possible given this attack relies on the physical properties of the RAM modules themselves. The researchers who found it said that greater quality control in RAM will help, as the interference between memory cells is a flaw, just one that has been allowed into RAM modules in the past. Also, that this works against the ECC RAM commonly used in servers seems to me to make it that much more dangerous.