Probably the last remnant in English of Germanic impersonal verbs, a common feature of many Germanic languages where the subject of many nonvolitional verbs of perception is in the dative case rather than the nominative.
It is fossilized now in a fixed expression, but the verb “thinks” here is actually a different verb from the modern verb “think”. This difference is very much alive in, say, Dutch, where one would say “Ik denk dat ...” for “I think that ...”, but “Me dunkt dat ...” with a different vowel for the same meaning as “methinks”, which denotes a less voluntary perception, an observation if one will.
It's not that dissimilar to “To me, it appears that ...” I suppose, with the key difference that the grammar does not demand another subject. It is simply “Methinks that ...”, not “Me, it thinks that ....
In modern English, I don't see any difference between "methinks" and "I think" it's just two ways of saying the same thing. If anyone sees a different shade of meaning between the two, what is it? And "To me, it appears that..." is just more words to say "I think..."
It is fossilized now in a fixed expression, but the verb “thinks” here is actually a different verb from the modern verb “think”. This difference is very much alive in, say, Dutch, where one would say “Ik denk dat ...” for “I think that ...”, but “Me dunkt dat ...” with a different vowel for the same meaning as “methinks”, which denotes a less voluntary perception, an observation if one will.
It's not that dissimilar to “To me, it appears that ...” I suppose, with the key difference that the grammar does not demand another subject. It is simply “Methinks that ...”, not “Me, it thinks that ....