Learn English – Why is it ‘come’ and not ‘comes’ in ‘it is important that he come…’

Short answer:

Because subjunctive mood obligates the following:

The present tense third person singular drops the -s or -es so that it looks and sounds like the present tense for everything else. — source.

Longer answer:

Well, many argue that the subjunctive mood is disappearing from English. Somewhat rightfully so, considering that other languages tend to have more sophisticated subjunctive systems.

How does subjunctive "work"? This is the same reason some people insist that you use

If I were able to swim, . . .

rather than

If I was able to swim, . . .

For most verbs, the only visible and distinct difference is the one you noticed: No -s or -es ending as otherwise would've existed. For verb 'be', we both have "be" and "were" subjunctives, which of course, are used in present and past tenses, respectively.

I urge that she meet Dr. Whom ASAP.

I think I clearly stated that I disagree with the piece of advice that he be the guy to confess to uncommitted crimes.

If I were you, I would've hired a consultant.

For the most part, indicative mood is subjunctive mood's counterpart.

When should I use subjunctive mood in modern English grammar? John Lawler has a notable post on when to use subjunctive mood in modern English, and more precisely, the patterns in which the subjunctive appears. To quote the answer:

There are four patterns, with four different kinds of complement-taking impositive predicates:

1) Transitive impositive communication verbs:
insist, suggest, demand, prefer, propose, suggest, recommend, demand, ask, mandate, prefer, request, ask, desire, advise, urge, specify, instruct, order, demand, insist, require, rule, necessitate, suffice, advocate, vote, would rather, and move (in the parliamentary sense).

Pattern: NPVolit Verb that [NP + Infinitive VP]

2) Transitive impositive emotive predicate adjective: adamant

Pattern: NPVolit be Adj that [NP + Inf VP]

3) Intransitive impositive predicate adjectives (normally with Extraposition):
necessary, desirable, imperative, important, necessary, preferable, optional, permissible, acceptable, okay, all right, satisfactory, desirable, advisable, sufficient, necessary, mandatory, urgent, vital, crucial, essential, fitting, right, appropriate, better, expedient, and legitimate

Pattern: That [NP + Inf VP] be Adj = Extraposition => It be Adj that [NP + Inf VP]

4) Picture nouns derived from impositive predicates:
recommendation, necessity, insistence, proposal, preference, request, desire, advice, suggestion, option, alternative, recommendation, demand, requirement, necessity, imperative, condition, mandate, specification, rule, ruling, edict, instruction, principle, prerequisite, order, qualification, ultimatum, vote, and motion (in the parliamentary sense).

Pattern: [PictureN that [NP + Inf VP]] (not a clause, but an NP with a complement clause)

_{Examples omitted to prevent from adding unnecessary length to the answer}

Back to your sentence,

It is important that he come to my party.

It uses the subjunctive mood and fits pattern 3.

And what does it mean? Various online sources for learning English agree that subjunctive mood refer to a conditional or doubtful situation. Subjunctive is an irrealis mood, meaning, it doesn't necessarily point to something real. He may come to the party, he may not. To quote Wikipedia

Subjunctive forms of verbs are typically used to express various states of unreality such as wish, emotion, possibility, judgment, opinion, obligation, or action that has not yet occurred; the precise situations in which they are used vary from language to language.
The English subjunctive is used to form sentences that do not describe known objective facts. These include statements about one's state of mind, such as opinion, belief, purpose, intention, or desire. — English Subjuctive, Wikipedia

So what the sentence means is "His coming into my party is important." This course of action (coming to the party) he will take may happen or not as it refers to the future, but it's important that he does it.