Imagine a graph which is progressively destroyed by cutting its edges one after the other in a uniform random order. The so-called cut-tree records key steps of this destruction process. It can be viewed as a random metric space equipped with a natural probability mass. In this work, we show that the cut-tree of a random recursive tree of size $n$, rescaled by the factor $n^{-1}\ln n$, converges in probability as $n\to \infty$ in the sense of Gromov-Hausdorff-Prokhorov, to the unit interval endowed with the usual distance and Lebesgue measure. This enables us to explain and extend some recent results of Kuba and Panholzer on multiple isolation of nodes in random recursive trees.