Typically it refers simply to having fore-thought of an occurrence, or one's expectation of something before it happens. Your example might be better phrased "The meeting had been anticipated since last week" or "I anticipated his every move in that chess game." In each case the sentence is referring to preparation, or precognition, of an event.
Typically anticipated will be tied together with a sentence like this "I anticipated his drive down the middle of the basketball court, so I planted my feet and took the foul."
Otherwise, the example could be tied to your question better if phrased as 'The meeting anticipated the budget shortfall.' In which case you are still dealing in the same sense and basic usage of normal "anticipate", but rather than a mental anticipation it is an event/object that is in advance of.