See the question Geometric interpretation of trace. There are several ways to define the trace:

• The sum of elements on the main diagonal.

• The sum of eigenvalues.

• The derivative of the determinant at the identity.

• The unique Lie algebra homomorphism onto $\mathbb{R}$, up to scale. (See also here.)

• "What you get when a linear map eats itself", as one answer put it. More precisely:

The idea of the trace operation is easily seen in string diagram notation: essentially one takes the endomorphism $a \stackrel{f}{\to} a$, "bends it around" using the duality and the symmetry and connects its output to its input.

This comment tells a story about it:

This reminds me of a story recounted by a friend of mine in graduate school. He spent a lot of time in the department, and one evening was approached by an undergraduate taking a fancy class that had introduced the trace of a linear transformation in the slick coordinate-free manner. This undergraduate had been tasked with computing the trace of a certain 2×2 matrix and had no idea how to proceed.

Nontrivial divides between coordinate-free/abstract and coordinate-based/concrete approaches—turning theorems into definitions and viceversa—also arise elsewhere, such as when defining tensors and tensor products.