Can anyone give a reference, a proof, or a reference that explains why Maple can evaluate this identity mathematically correctly:

$$n-i-1=(d-1)\sum_{l=1}^{n-i-1}\frac{\binom{n-i-1}{l}}{\binom{n-i+d-3}{l}}$$

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The canonical reference for this sort of thing is Petkovsek and Zeilberger's book "A=B". Maple (almost certainly) uses the Zeilberger-Wilf algorithm for hypergeometric summation (which really goes back to Bill Gosper). You can also read the Wilf-Zeilberger paper (Inventiones, around 1990).

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As Gerhard Paseman suggested, it look better to replace $m=n-i-1$ and also $x=d-2$. With this the question takes the form $S:=\sum_{\ell=1}^m\binom{m}{\ell}\binom{m+x}{\ell}^{-1}\frac{x+1}m$ and we show $S\equiv1$. Let $$F(m,\ell):=\binom{m}{\ell}\binom{m+x}{\ell}^{-1}\frac{x+1}m \qquad \text{and} \qquad G(m,\ell):=-\binom{m}{\ell}\binom{m+x+1}{\ell}^{-1}\frac{m+x+1}m.$$ It's easy to check $$F(m,\ell)=G(m,\ell+1)-G(m,\ell), \tag1$$ say by dividing through with $F(m,\ell)$ and simplifying. Summing (1) over integers $\ell\geq0$ gives $$\sum_{\ell=0}^mF(m,\ell) =\sum_{\ell\geq0}G(m,\ell+1)-\sum_{\ell\geq0}G(m,\ell)=-G(m,0)=\frac{x+m+1}m.$$ Therefore, we arrive at $$S=\sum_{\ell=1}^mF(m,\ell)=\sum_{\ell=0}^mF(m,\ell)-F(m,0)=\frac{x+m+1}m-\frac{x+1}m=1$$ which is what aimed for.

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or
why, are you askinghowdoes Maple derive the identity. $$ $$ Anyway, the question is not clear as is, it needs motivation. See: mathoverflow.net/howtoask $\endgroup$1more comment