When you rotate the tires on your car how does that happen? Well you take at least two off and swap them yes?
Bit storage be it individual flip-flops or gobzillions of them packed in a ram. Each bit of storage is readable and writeable. otherwise what is the point? A bit shift is nothing more than an operation that takes the tires off, reads the bits out of a register or memory location, and then writes them back in a different location, just like moving one tire from the back to the front, you read it out, then write it back in a different location.
There is no magic in logic, it is all very very simple in concept. zero, one, and, or, not is all that is required to understand. It is just at times a massive amount of and, ors and nots in parallel and/or series.
Some processors shift one bit at a time and in pseudo hdl it is not really different than in software
alu_out = (operand_a[14:0],0)
the 16 bit output is made of the lower 15 bits of the input re-positioned to the upper 15 bits of the output with a zero tacked on in the 0 position.
And that line of code may be in a switch like statement or tree of if then else statements that basically says if alu_operation = left_shift then do this.
And all of that is compiled into logic gates, the ifs and the assignments turn in to and, or, not gates or gates derived from and, or, not. on a massive rats nest.
Some processors have a variable sized shifter that in one operation can rotate or shift however much you want (up to the whole word size obviously, otherwise what is the point) granted that takes a lot more gates, just that one operation each bit of the output for a 16 bit register has a 15, 16, or 17 to one mux depending, just for that one operation.