What init 1 command exactly does?

ID  Name                                   Description
0   Halt                                   Shuts down the system.
1   Single-user mode                       Mode for administrative tasks.
2   Multi-user mode                        Does not configure network interfaces and does not export networks services.
3   Multi-user mode with networking        Starts the system normally.
4   Not used/user-definable                For special purposes.
5   Start the system normally with         Same as runlevel 3 + display manager.
    appropriate display manager (with GUI)                                         
6   Reboot                                 Reboots the system.

As you can see from the above each runlevel has what commands and programs run when started by user or any other process, as in the case of runlevel 1. Here only a single user can access the system and the commands here are barest at minimum, hence the comment that it freezes the system. Most commands you know in the 3-5 runlevel are not available here.

If you try entering this level from your current runlevel, you will see that a lot of things will not work and you most likely lose data in any open files or programs. Best tried on a virtual machine to see what it does.

Basically in Linux every runlevel is associated with certain programs or programs are associated with various runlevels. So entering that level with init 1 will kill all programs that are not designed or meant to be in that runlevel, hence the computer freeze mentioned by you.

Source: https://en.wikipedia.org/wiki/Runlevel

I am not a Linux programmer nor am I a hacker but I too have been watching Mr. Robot and came here for answer.

The line in the TV show was:

Darlene: "My computer kept crashing, and you (Elliot) told me to try init1"

Now I am a c++ programmer and electrical engineer so I checked online and got my answers here. Very simply it was because Darlene was using a higher runlevel and her computer would crash consistently being the cause of her frustrations. Elliot suggested to run the single user mode thus eliminating the networking and other users and allowing her to debug the program.

init1 became their codeword whenever she needed help.