Why is it called linearly independent?

An expression of the form $a_1V_1 + a_2V_2 + ... + a_nV_n$ for vectors $V_i$ and scalars $a_i$ is called a "linear combination" because it generalizes the property that $y = ax$ is a line through the origin in the plane. It is simply a description that someone attached to the concept in the past, and is now common use.

A set of non-zero vectors are linearly dependent if at least one of them can be written as a linear combination of the others. It depends on the others. As such, you could kick that vector out as redundant, as everything that is a linear combination of the full set is also a linear combination of the other vectors without it.

A set of vectors is linearly independent if none of its vectors is a linear combination of the others. In this set, there are no redundant vectors. Throw out one, and you get a smaller span.

We have linear which is self-explanatory - 'of lines', and independence which means not reliant on each other, and the dictionary.com references the archaic definition competence.

So a linearly independent set of vectors is a set of lines that competently (necessary and sufficient) defines a vector space.

I mean, what are they independent of?

They are independent of each other.