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I built a network dataset (a system of roads) and loaded multiple facilities in the "Facilities" layer of the NA window.

I clicked "Solve" to run the analysis and saw that my network lines now have calculated distances.

I wonder, what are these distances?

The online documentation is unclear. Under "Multiple Facilities Option", there is an explanation of how the tool handles polygons from multiple facilities, but no mention of how it handles the ToCumul_Length column created in the resulting service area.

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In the service area analysis of NA, how the line features (if you ask to be generated) are generated/calculated is well-explained in the link that you have given. ToCumul_ (impedance is the length in your case) and FromCumul_... fields give how much distance is traversed based on the facilities that you have inputted (please see Output fields of lines table under Line properties section of the documentation for the difference in calculation).

Most probably your confusion is related to the overlapping lines belonging to more than one facility's service areas. If you look at the The Line Generation tab section of the aforementioned documentation, under Overlap Options part says, if you select overlapping lines option, it will give you the least cost/impedance route (you can pick up which facility is that from SourceID, SourceOID fields) among all possible options. Otherwise, tool creates individual sets of lines for each facility. Again, as a rule of thumb, if there are overlapping lines within the service area of one facility, you will be given the least costly one.

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