What exactly does this option do? I've been reading a lot on TTY and I'm still confused. I played around without having the -t and just -i and it seems like programs that expect user input throw an error without the -t. Why is it important for pseudo-TTY to be enabled?
The -t option goes to how Unix/Linux handles terminal access. In the past, a terminal was a hardline connection, later a modem based connection. These had physical device drivers (they were real pieces of equipment). Once generalized networks came into use, a pseudo-terminal driver was developed. This is because it creates a separation between understanding what terminal capabilities can be used without the need to write it into your program directly (read man pages on stty, curses).
So, with that as background, run a container with no options and by default you have a stdout stream (so docker run | <cmd> works); run with -i, and you get stdin stream added (so <cmd> | docker run -i works); use -t, usually in the combination -it and you have a terminal driver added, which if you are interacting with the process is likely what you want. It basically makes the container start look like a terminal connection session.