Most would define inspired as something like ‘animated’ or ‘excited’ but, as is the case with so many modern words, these definitions and associations focus on the result and not the root cause. They fail to address the question of what is providing the force that is driving the animation or excitement being observed.
As always with a multisyllabic word, the best way to understand it is to break it down into its constituent parts, examine and define them, and then build the word back up from its constituent parts.
Inspire is constructed from the prefix in- meaning that something is ‘taken in’ or ‘consumed’ (think of ‘insert’ or in slightly altered form, ‘imbibe’) into some type of larger body, often a human one.
The second half of the word, -spire, comes from the Latin word spirare which means ‘to breathe’. So, read literally, the original meaning of ‘inspire’ is ‘to breathe in’ or ‘to take in a breath’.
At this point, though, we are still dealing with a purely materialistic concept, the act of respiration (this word being another illustration of the same theme). What we are missing in this functional understanding of the word is what the concept of ‘breath’ signifies in terms of the religious and esoteric thought that would have permeated the ancient world in which the word was formed: divinely delivered creative life energy.
For example, in the Völuspá it is Odin, the god of poetry and ecstasy, who provides man with breath, while in Judeo-Christian tradition God does the same for Adam. Further afield, H.P. Blavatsky refers in The Secret Doctrine to the esoteric concept of vibration as “the thrill of the creative Breath in Nature.”
All of these references point to the final connection we need to make in order to understand the true meaning of the word which, as Blavatsky points out explicitly, is related to receiving the breath of a god that imbues the recipient with creative powers.
While this is most likely not uppermost in the minds of modern people when they use the word inspired, I believe that it is subconsciously understood due to the way the word is used, which is almost always in contexts related to creative endeavor, whether artistic or of some other variety.