Meltdown (as well as Specter) has recently been released to the public, and there are a lot of long and confusing articles that talk about memory management, speculative execution, timers, arrays, etc, but that can all be very confusing. Here is an article that describes how this really works in terms that you can understand.
One of the efficiencies that modern processors employ is the use of multiple realities. Sometimes, your processor performs execution instructions before privilege checking can occur. If the privilege check is successful, it has the result of the instruction ready for use. If the privilege check fails, then the processor discards the result, and it is never committed to any register or memory space (a register is just a very small piece of memory that lives in your processor); HOWEVER, in an alternate reality, your the privilege check was SUCCESSFUL, and the value of the execution WAS committed somewhere. If a process can access this alternate reality, it can gain access to the result of the execution that failed the privilege check in this reality. If this execution was requesting a piece of memory which was restricted (failed privilege check), the special quantum process can access the reality where this check succeeded, thus obtaining secret contents from memory. See below for an illustration of this process.
TL;DR - Processors are so fast that they induce a type of quantum superposition regarding their execution results, and can be read by a process that lives in different realities where the outcome opposite of this reality was successful.