I think that the dative sibi can be crucially related to the presence of the preverb con-. A typical use of the dative case is found with some compound verbs and adjectives.
sī sibi ipse cōnsentit (Cic. Off. I.5) 'if he is in accord with himself'
So note that the the same dative you see with adjective conscius also appears with the related verb conscire:
nil conscire sibi (Hor. Ep. 1, 1, 61) lit. 'to be conscious with himself of nothing', i.e., 'to be conscious of no guilt'
It can also be useful for you to take a look at what is summarized in the wikipedia entry of consciousness. Here is a copy&paste of the relevant fragment:
"The earliest English language uses of "conscious" and "consciousness" date back, however, to the 1500s. The English word "conscious" originally derived from the Latin conscius (con- 'together' and scio 'to know'), but the Latin word did not have the same meaning as our word —it meant "knowing with", in other words "having joint or common knowledge with another". There were, however, many occurrences in Latin writings of the phrase conscius sibi, which translates literally as "knowing with oneself", or in other words "sharing knowledge with oneself about something". This phrase had the figurative meaning of "knowing that one knows", as the modern English word "conscious" does. In its earliest uses in the 1500s, the English word "conscious" retained the meaning of the Latin conscius. For example, Thomas Hobbes in Leviathan wrote: "Where two, or more men, know of one and the same fact, they are said to be Conscious of it one to another." The Latin phrase conscius sibi, whose meaning was more closely related to the current concept of consciousness, was rendered in English as "conscious to oneself" or "conscious unto oneself". For example, Archbishop Ussher wrote in 1613 of "being so conscious unto myself of my great weakness". Locke's definition from 1690 illustrates that a gradual shift in meaning had taken place".
Finally, if you're interested in philosophical issues, you can also take a look at this link.