It looks like-- and means-- 'I think', but it has come directly and almost unchanged from the Anglo-Saxon 'it seems to me'. 'Me' instead of 'I' because it is a remnant of Germanic dative case (used for indirect object). '-S' of third person singular because the subject of the old verb (see below) is 'it'.
'Methinks, v. impers.
[AS. şyncan to seem, me şynceğ, me şuhte, OE. me thinketh, me thoughte; akin to G. dünken to seem, denken to think, and E. think.]'