"Into" is a preposition that indicates entering a space. (Most prepositions are directions like the arrows on a flowchart, expressing towards/from/into/around/over/under something.) Prepositions take a noun as their object. For example, "I'm going INTO the store," or "I'm throwing a ball INTO the bucket."
If you say "I'm going in to see someone," the "to" is being used not as the preposition "to" meaning a destination ("to London"), but rather, as the first half of what's called an infinitive, a special form of the verb "to run, to walk, to fly."
In that case, I believe, the "in" is NOT a preposition; it's acting as an adverb describing the verb. Adverbs answer questions like when/where/how.
I'm not sure if all of that makes sense. Basically, if the word that follows "into" is a noun, no space; if it's a verb, then a space is okay.
English is confusing because so many of our little words like in and to do double and triple duty, acting as different parts of speech in different phrases and usages.