Why is shift register very important?
The applications of shift registers makes it important for microcontrollers. A few of the applications of shift registers are:

'Serial to Parallel' and 'Parallel to Serial' Conversion
Usually microprocessors and controllers handle data as bytes (8 bits) or words (16 bit, 32 bit .. ). But serial interfaces in them allow transmitting or receiving data as a series of bits (serial data). So there should be a mechanism that will convert the serial data into parallel data (or vice versa). The Serial-In-Parallel-Out (SIPO) and Parallel-in-Serial-Out (PISO) shift registers makes this possible.

Multipliers
The basic steps involved in binary multiplication is shifting and adding. Hence shift registers are a part of binary multipliers. 

Registers to store data
Used in ALU's to store operands, intermediate results and final results.