Binding Of Isaac Rebirth, Red Icon

You are playing on Hard, probably selected it inadvertently on the character select screen. It's easy to choose it accidentally because it's selected via the up/down arrows/sticks.

The red icon means your playing in hard mode.

It is very easy to mis-click the button and go to hard mode, so you have to be on the look out if your not ready for the immense difficulty.

Why people play on Hard Mode:

There is a "hard mode" for more experienced players, giving away a few new rewards that are unattainable in other difficulties.

What is different in the Hard Mode vs. Normal Mode:

1. Beating different goals in Hard mode will unlock new content not unlockable on Normal.

2. Beating said goals in Hard mode will also unlock any unlocks that would be gained by beating Normal mode (so you can start on hard if you’d like)

3. There will be achievements for Hard mode.

4. Hard mode will have some visual hud/menu changes so you are always aware if you are playing on Hard. (or watching someone play on Hard)

You can see where you click to get into Hard Mode here:

Source

Source 2