分かる is typically translated as “to understand” in English, but more literally, it’s something like “to be understandable/to be understood.”
So the subject is indeed the thing being understood –
日本語が分かる is literally “Japanese is understood,” and context tells you who is doing the understanding.
But since that sounds weird in English, we translate it as “I understand Japanese.”
In case it’s necessary to specify who’s doing the understanding, you can add that information with は.
Example:
田中さんは日本語が分かる。
“As for Tanaka-san, Japanese is understandable (to him).”
Or in more natural English, “Tanaka-san understands Japanese.”
(は is not a subject marker but a topic marker, hence the “as for” translation.)
Also, just to be clear, this is unrelated to it being the plain-form, the same would be true of わかります.
There are several verbs like this where Japanese has something be the subject that the equivalent English verb puts as the object. This set of examples is from the Dictionary of Basic Japanese Grammar:
私は車がある. – lit “as for me, a car exists” → I have a car
僕はお金がいる – lit. “as for me, money is necessary” → I need money
私はフランス語が少し出来る – lit. “as for me, French is a little bit possible” → I can speak a little French
ここから富士山が見えますよ – lit. “from here, Mount Fuji is visible” → You can see Fuji from here
There are also verbs where English puts something as a direct object but in Japanese it’s an indirect object (marked with に) or it’s marked with と.
I think the best thing with this kind of verb is not to get too hung up over what the English translation is – the verbs themselves in Japanese work the same way other verbs do, there just isn’t any English verb that exactly corresponds to them.
You’ll find 見える at WK level 5 (“to be visible”) and 聞こえる at level 12 (“to be hearable”). These give some people difficulty for similar reasons: they’re ordinary (and quite common) intransitive Japanese verbs, but the natural way to phrase the idea in English is often with the potential (“I can see/hear ~”).
I feel like if we are throwing potentials into the mix it’s worth just outright saying that potential verbs (can XYZ) typically just use が as a feature of Japanese.
Just to be clear in this particular case, 見える is not the potential form of 見る, even though it looks like it might be. That’s because 見る is an ichi-dan verb. It’s only the go-dan verbs that get a simple え inserted for the potential form. The potential form of 見る is 見られる (able to see). 見える’s potential form is 見えられる (able to be visible).
Unfortunately, the potential form of ichi-dan verbs is identical to their passive form, so you really need to watch the particles to figure out what’s what.
Thanks for the clarification. Yeah, I should have been a bit more clear that when I said “potential verbs” I wasn’t referring only to “verbs in potential form” but also to verbs where potentiality (is that a word?) is part of their definition. 見える, for example, is one of those words. Which is why you won’t see it in potential as 見えられる