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You should read $c_1(X)<0$ as saying that the first Chern class of $T_X$ is negative, or the line bundle $K_X$ is positive, in the sense of curvature. But positive line bundles correspond to are ample line bundles. This correspondence fact is sometimes called the Kodaira Embedding Theorem. See for example p. 181 of Griffiths-Harris, Principles of Algebraic Geometry.

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You should read $c_1(X)<0$ as saying that the first Chern class of $T_X$ is negative, or the line bundle $K_X$ is positive, in the sense of curvature. But positive line bundles correspond to ample line bundles. This correspondence is sometimes called the Kodaira Embedding Theorem. See for example p. 181 of Griffiths-Harris, Principles of Algebraic Geometry.