For the verb puto -are, Cassell's and Lewis & Short give a primary meaning of "to clean, cleanse" and a literal meaning of "to trim, prune or lop trees or vines". I can easily imagine the path that led from that to the figurative sense of clearing up, setting in order, or arranging, settling and adjusting. But the image is not so clear when the usage becomes more generalised, when the verb starts being applied to a wide range of cognitive processes. The dictionary entries give not only specific processes like judging, esteeming, computing, accounting and suspecting as legitimate readings, but also broader processes like thinking, imagining, supposing and believing. To me, this relation between "cleansing, pruning, trimming" and "judging, reckoning, thinking, etc." is not obvious. I assume it was obvious to the Romans, so I am sure my failure in making a connection is just due to a lapse of imagination (and a lack of reading) on my part. At any rate, how do we account for this development? Do the Romans themselves have any explanations to offer?
Since this question received no answers in more than a month, let me make a feeble attempt at coming up with one. When we enter the realm of semantics, it is easy to concoct an ex post explanation, a just-so story; theories are scarce though. But let me start with the quote from the OP:
But the image is not so clear when the usage becomes more generalised, when the verb starts being applied to a wide range of cognitive processes.
The boldface is mine. Is not it clear how the semantics of clearing up a space or view (say by trimming trees) translates into that of mental clarity?
It is a common pattern, at the least in the IE languages, when the terms of clearing out space translate metaphorically into those of mental clarity, and those of understanding are from grasping (grasping in the sense of understanding, as also in Rus. понять ponyat' = “to understand, to realize”, literally ,“to seize-PERF”), or reaching a locus in space as the end of a thinking "path" (both understand = “stand under” and ἐπίστᾰμαι = “stand over/upon” are clear examples of this pattern). Yes, we reach a conclusion. And the conclusion itself is from con-claudere, something we lock up and make immovable. The metaphor is but the most productive vehicle for the creation of new concepts.
As for @Cerberus' additional question, the words of decision come from the root meaninig “cut” in so many IE languages as well. decide = cut off. Rus. решить reshyt' = to cut. Resolve also contains the PIE root of *leu- with the similar semantics of cutting. This is another common theme.
I wish I could come up with a more scientific explanation, but there is no theory on how mind produces language worth calling scientific, and not even one of semantics development only. I can only share some partial generalizations I came across as important in my field of research.
Perhaps, an accessible and popular book emphasizing the workings of the metaphor, among other topics of language development (many of them fascinating, a few possibly controversial), would be The Unfolding of Language by Guy Deutscher.