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This should be is a comment to Charles's answer, but I can't yet leave need more room than what comments . (I am a newbie)allow.

Anyway, what "glue back differently" means is that the curve is "glued back" with its normal bundle "reversed".

There is also an algebraic way to think about flips: If $f:X\to Y$ is a contraction, then $X$ can be considered as ${\rm Proj}_Y\sum_{m=0}^\infty f_*\mathcal O_X(-mK_X)$. Now if $f$ is small, then the flip of $f$ is given by the morphism $f^+: X^+={\rm Proj}_Y\sum_{m=0}^\infty f_*\mathcal O_X(mK_X)\to Y$. So, to prove the existence of a flip you "only" need to prove that the above algebra is finitely generated over $\mathcal O_Y$.

This might not seem an intuitive way right away, but remember that Proj comes with a relatively ample divisor, so what's happening is that we make an $f$-anti-ample divisor into an $f^+$-ample one without changing it on the locus where $f$ was an isomorphism. If $X$ and $Y$ are $3$-dimensional and $f$ is a small Mori-contraction then it contracts a single rational curve and being ample is equivalent to the degree of the divisor on the curve being positive. Now the (anti-)ampleness of the canonical class is then governed by the normal bundle of the curve and hence "flipping" the positivity of $K_X$ on this curve is essentially the same as "flipping" the normal bundle.
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This should be a comment to Charles's answer, but I can't yet leave comments. (I am a newbie).

Anyway, what "glue back differently" means is that the curve is "glued back" with its normal bundle "reversed".

There is also an algebraic way to think about flips: If $f:X\to Y$ is a contraction, then $X$ can be considered as ${\rm Proj}_Y\sum_{m=0}^\infty f_*\mathcal O_X(-mK_X)$. Now if $f$ is small, then the flip of $f$ is given by the morphism $f^+: X^+={\rm Proj}_Y\sum_{m=0}^\infty f_*\mathcal O_X(mK_X)\to Y$. So, to prove the existence of a flip you "only" need to prove that the above algebra is finitely generated over $\mathcal O_Y$.

This might not seem an intuitive way right away, but remember that Proj comes with a relatively ample divisor, so what's happening is that we make an $f$-anti-ample divisor into an $f^+$-ample one without changing it on the locus where $f$ was an isomorphism. If $X$ and $Y$ are $3$-dimensional and $f$ is a small Mori-contraction then it contracts a single rational curve and being ample is equivalent to the degree of the divisor on the curve being positive. Now the (anti-)ampleness of the canonical class is then governed by the normal bundle of the curve and hence "flipping" the positivity of $K_X$ on this curve is essentially the same as "flipping" the normal bundle.