How is an Assembly Language Processed by a CPU's Circuitry?
An assembler is a program that reads assembly language commands and translates then into a sequence of binary instructions, addresses and data values that is called machine code. The machine code is stored in the computer's memory and can be executed by the computer at some later time. Machine code is read and "understood" directly by the CPU.

So a command such as "load the value 10 into register A" might be written in assembly language as "LDA 10" and then stored in machine code as one byte 00101010. The first four bits of the machine code instruction 0010 represent the LDA instruction and the second four bits 1010 represent the value that is to be loaded.

Note that the assembler makes life easier for the programmer by translating the "LDA" instruction and translating the value 10 from decimal to binary. It will also do other stuff like allowing the programmer to use labels, which it then translates into specific memory addresses.

In the simplest CPU architecture, when the CPU executes the instruction 00101010 it actually runs a sequence of low level microinstructions which will be something like this:


Add 1 to the register that tells the CPU where the next instruction is stored in memory 
Set a control line to take control of the data bus.
Load the lowest four bits of the machine code instruction onto the data bus.
Release control of the data bus.
Set a control line to tell Register A to read and store the value on the data bus.
Read the next instruction from memory.


In the very simplest/oldest CPU architecture this final translation from machine code to microinstructions is hard wired in logic gates.

A good guide to this sort of stuff for beginners is "But How Do It Know".

Bonus question: If an assembler is a program that creates other executable programs, how is the assembler created in the first place ?