The best way to answer these questions is to run bedtools cluster yourself and see what happens. That said I was able to surmise the following from the documentation:

the default output is a three column BED file with a 4th column denoting which cluster the feature has been grouped in, see https://bedtools.readthedocs.io/en/latest/content/tools/cluster.html#default-behavior for the example.

Since there is no flag for specifying the name of the output file I imagine the output is written to standard output and can be written to file by adding > cluster_output.txt to the end of you command.