how do we know about parts of Buddha's life where he was clearly alone?
Some of it is in suttas, see for example Buddha meditation under the bodhi tree

The detail you mentioned come from the introduction to Jataka -- Where is the description of the vow and the Bodhi tree? 

Parts of the Jataka might be described here as "a charming story" -- see also Does Theravada Buddhism accept Jataka Stories?

See also Why did the Buddha touch the earth at his enlightenment? -- people understand that as instructive.