[I'd already typed this long version before ydtt's reply, so I'll post it anyway.]
皆に is a dative subject, not an indirect object, because みえる is a stative verb.
One test to identify the subject is honorifics: 先生に 日本語が おわかりになります。
About 'stative': all adjectives (na and i) are stative b/c they describe a property or state of the subject rather an action or event. Most verbs describe actions or events, but there are a few which describe a property. They're called 'stative verbs'. All verbs in their －ている form are stative too because they describe a state.
I think DoBJG has a list of stative verbs. Common ones include: existence/possesion (いる、ある), ability (わかる、potential verbs), need (いる), perception (見える、聞こえる）. These are non-volitional verbs when used as stative verbs (some also have non-stative meanings)
The object of these stative verbs (the thing which is possessed, understandable, visible, needed, etc) is marked with が and sometimes を. (You've probably encountered that alternation with potential verbs.). The subject (the person experiencing or possessing) is marked with に or が. Marking the subject of a stative predicate with が gives it focus, though, unless it's in a relative clause. ('It is BOB who can understand Swahili.') A dative (に) subject never occurs with を object.
These stative verb constructions are neither pure transitive nor intransitive. The verb might be labelled 'intransitive' in the dictionary, but intransitive verbs are only supposed to have one participant. Stative verbs, however, require the existence of an experiencer person as well so there are two participants. Better to just learn them as stative constructions - another hybrid category.
There aren't that many of them and the predicate vs attributive forms of a few of them are restricted which you might want to know at some point.
日本語がわかる
It doesn't make much sense to say 'Japanese is understandable'. It's a language after all. There must be an unspoken person who can understand it (often 私).
彼に 日本語が わかる
彼が 日本語が わかる（こと）
(こと makes it a neutral subject, not focused, which apparently can sound awkward)
彼が 日本語を わかる (less common, apparently more by younger people?
He can understand Japanese. Japanese is understandable to him.
If the subject is a topic, it's either 彼は or 彼には
* * *
About verbs vs i adj:
As ydtt said, Japanese and English parts of speech don't match up perfectly. For eg, so-called na-adjectives have properties of both nouns and adjectives. So shall we call them 'nominal adjectives' or 'adjectival nouns'? Similarly, i-adjectives are more verb-like than na-adjectives, but they don't have all the same properties as verbs. (different conjugation, don't combine with aux verbs, no formal conjugation, no imperative, etc.) So I suppose we could call them 'verbal adjectives' or 'adjectival verbs'? But the latter would be confusing b/c it's the literal English translation of 形容動詞 (in traditional Japanese grammar だ was considered part of those words.)
It gets described as a continuum from most noun-like to most verb-like:
Noun........na adj..........i adj.......verb
One suggested explanation of the difference bw verbs and i adj is that the い in i-adj is a kind of copula-ish (だ) thing that evolved the way ydtt described. So it's like adj + だ, but different. ;p
btw, English also has verbs acting like adjectives: The bear is dancing. The dancing bear.
