"The Elder Wand cannot serve me properly, Severus, because I am not
its true master. The Elder Wand belongs to the wizard who killed its
last owner. You killed Albus Dumbledore. While you live, Severus, the
Elder Wand cannot truly be mine."
"My Lord!" Snape protested, raising his wand.
"It cannot be any other way," said Voldemort. "I must master the wand,
Severus. Master the wand, and I master Potter at last."
And Voldemort swiped the air with the Elder Wand. It did nothing to
Snape, who for a split second seemed to think he had been reprieved:
but then Voldemort's intention became clear. The snake's cage was
rolling through the air, and before Snape could do anything more than
yell, it had encased him, head and shoulders, and Voldemort spoke in
Parseltongue.
"Kill."
There was a terrible scream. Harry saw Snape's face losing the little
color it had left; it whitened as his black eyes widened, as the
snake's fangs pierced his neck, as he failed to push the enchanted
cage off himself, as his knees gave way and he fell to the floor.
We can see from the books, that a few different stuff happened leading to Snape dying:
Snape did try to protect himself. He raised his wand once Voldemort's intentions were clear. He didn't immediately attack- Maybe because he thought he can talk his way out of it, or because he preferred a strategy of defending himself rather than attacking the dark lord.
Snape got confused- He saw Voldemort casting a spell, and then nothing happened. He felt falsely secure for a moment and thought he got away with it.
Then Voldemort suddenly attacks, with the cage rolling to Snape, and Snape, caught by surprise, could barely yell before he gets stuck with Nagini in the magical cage. It's not obvious that Snape would protect himself immediately with magic- He instead starts screaming and instinctively tries to push the cage off him as the snake kills him.
As for not disapparating- I have reason to believe that a certain Voldemort wouldn't really want people to have easily free access to his lair with one of his last Horcruxes. To support this theory, we see that both Lucius Malfoy, who is urged to find Snape, and Voldemort, who should consider himself above crawling in a muddy tunnel for half an hour, WALK out of the room, instead of disaparating.
Frightened, stumbling a little through the gloom, Lucius left the
room.
He turned away; there was no sadness in him, no remorse. It was time
to leave this shack and take charge, with a wand that would now do his
full bidding. He pointed it at the starry cage holding the snake,
which drifted upward, off Snape, who fell sideways onto the floor,
blood gushing from the wounds in his neck. Voldemort swept from the
room without a backward glance, and the great serpent floated after
him in its huge protective sphere.
