This is exercise 18.4.J in Vakil's "The Rising Sea Foundations of Algebraic Geometry". Here the degree of a coherent sheaf $\mathscr{F} $ on integral projective curve $ C $ is defined to be $$ \mathrm{deg} \mathscr{F}=\chi(C,\mathscr{F})-(\mathrm{rank} \mathscr{F})\cdot \chi(C,\mathscr{O}_C).$$
Here the curve may be singular, and thus the local ring at a closed point is not a discrete valuation ring. I have no idea whether we can define the order of zeros or poles of a rational section in this case.
A hint is given: Exercise 13.5.H, i.e., given an exact sequence $$0\to \mathscr{F}_1\to \cdots \to \mathscr{F}_n \to 0 $$ of finite rank locally free sheaves on $ X $, the alternating product of determinant bundle is trivial.
My idea is that since $\mathscr{F}(m) $ is gloally generated for $m>>0 $, we get an exact sequence of locally free sheaf $$ 0\to \mathscr{F_0} \to \oplus\mathscr{O}(-m)\to \mathscr{F}\to 0,$$ but I don't know whether $\mathscr{F_0}$ is direct sum of line bundles, and I'm not sure if we still have $$\mathrm{deg}(\mathscr{F}\otimes\mathscr{G})=\mathrm{deg} \mathscr{F}+\mathrm{deg} \mathscr{G}$$ in this case. When C is regular, this is done by computing the divisor of zeros and poles of a rational section, but here $ C $ may be singular now.
Let $\mathcal{F}$ be a rank $n$ vector bundle (i.e. locally free sheaf) on such a curve $C$. Note that $\det(\mathcal{F})$ is a locally free sheaf of rank 1 (an invertible sheaf). We want to show that $\deg \mathcal{F} = \deg (\det \mathcal{F} ) $, i.e. $$ \chi(C, \mathcal{F}) - n \cdot \chi(C, \mathcal{O}_C) = \chi(C, \det(\mathcal{F})) - \chi(C, \mathcal{O}_C), $$ or equivalently, $$ \chi(C, \mathcal{F}) = \chi(C, \det(\mathcal{F})) + (n-1) \chi(C, \mathcal{O}_C). $$ Recall that Euler characteristic is additive in exact sequences, so we would be done if we can find some exact sequence $$ 0 \to ? \to \mathcal{F} \to \det ( \mathcal{F} ) \to 0 $$ or $$ 0 \to \det ( \mathcal{F} ) \to \mathcal{F} \to ? \to 0 $$ where $\chi(C, ?) = (n-1) \chi(C, \mathcal{O}_C)$. Thinking about the definition of $\det{\mathcal{F}}$, can you find such a sequence? The hint given in the text might give you some guidance but Vakil's hints can also be distracting at times.
Another route: use a result like https://stacks.math.columbia.edu/tag/0AYT, bullet point (4).