the backspace character

In the following char array, notice the use of a backspace character in a string literal: '\b'.

char text1[50] = "\aHello,\n\tWorld! Mistakee\b was \"Extra 'e'\"!\n";

What exactly does a backspace character do here? When the compiler evaluates this line, does it actually delete the previous character, like when you press the backspace button on the keyboard?
No. The compiler just inserts ASCII code 0x08 into the string, which "represents" a backspace. Terminals may interpret that as moving the caret back one space when writing out the string, but I don't think that is a requirement.
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