The red blurs are solar prominences which are features of the Sun that erupt from its surface and extend into the corona.

Staring at a lunar eclipse won't damage your eyes at all since the light from the moon is reflected sunlight. You can watch all phases of a lunar eclipse without any eye protection.

On the other hand, staring at the Sun with your naked eye during most phases of a solar eclipse (except totality) can be very dangerous since the light from even a sliver of the Sun in the partial phase can cause damage to your eyes. During totality -when the Moon completely covers the Sun's face- it is perfectly safe to watch without any eye protection.