Why does this sentences has "did" on it instead of "seemed"

It's stressing this thought. "You know, now that I think about it in the light of day, he really did seem to like me."

To elaborate on what other people have said, the difference is that "He seemed to like Madison" reads as a general observation, whereas "He did seem to like Madison" reads as acknowledging the observation after some deliberation.

the use of ‘did seem’ instead of ‘seemed’ puts emphasis on the ‘did’, for example implying that they were actively wondering whether or not he did, just realizing that he did in fact like madison right in that moment, or pointing it out to someone else

Same as in "He does have a point" instead of just saying "he has a point." Placing emphasis.

So its an emphasis.  Like if Person A (narrator) was talking to Person B (you) about Person C. “He’s been acting all funny lately at school. Well, he *did* seem to like Madison, so that might explain it.” Im sure the more classically educated English Speakers can get into more depth, but basically it is just a quirk of the language indicating a slightly different meaning than “He seemed to like Madison” Seemed has no real underlying tone Did seem - indicates a sort of musing to oneself or an outside person 

It's a form of emphasis typically used in spoken English. It can be used in narrative writing as a way of making the reader feel like the narrator is right there, talking to the reader.

Emphasis.

Emphasis on the auxiliary verb "seem". If you were to write it as "He seemed to like Madison," notice how that turned out to be a simple statement.

Emphasis

Everyone is just saying "emphasis", but that doesn't fully explain it. It often emphasizes that something else may have led you to believe that he *didn't* like Madison. For example: "He was always strict with her, but he *did* seem to like Madison."

This usage can also contrast it with before. Maybe he didn't think he'd like Madison but now that he got to know Madison better, he does, in fact, like Madison.

It's one application of do-support; as the others mentioned, here, it emphasizes the action, as the others mentioned. Do-support in general is a common thing, so you won't struggle with the rest of its usage (questions, tag questions, negations)