
'Methinks' is just an example of a word changing the way it's used. In current speech, we can say 'it occurs to me'. In the case of 'methinks', the
thinks part
is 3rd person. Similarly it was possible to say in Shakespeare's time (and he didn't, incidentally, use Old English) - where we would say 'I don't like it' -'It
likes me not'. The grammar of verbs of thinking, liking ... and other words dealing with states of mind, just changed.
A similar change happens across languages (both modern):
No me gusta la cerveza. - Spanish: the 'like' verb is in the 3rd person
Não gosto da cerveja - Portuguese: the 'like' verb is in the 1st person
b