An absolute phrase is a phrase that modifies/describes a noun within the sentence but is not connected to the rest of the sentence by a conjunction.
In your example, being wary describes the traveler and does not use a conjunction (because, and, so, but, yet . . .).
There are two verbs in your sentence. The main verb is sat (to sit, past tense). The second verb is being (to be, present tense) and it is part of the phrase that describes the traveler, not saying what the traveler is doing.
Contrast this with:
The traveler was weary.
The traveler sat by the wood side to rest because she was weary.
In these examples, the verb to be is functioning as the main verb or is more strongly joined to the rest of the sentence through the conjunction because.