Suppose you hire an agent to do a job. The agent's level of production p depends on the agent's effort m and on a mean-zero shock from nature s. So p = m + s. You observe p, but you do not observe m or s. Your agent observes s and p only after he has chosen m.
This is a standard principal-agent problem. If p is lower than you wanted, you cannot tell whether this is because your agent put in less effort than you wanted or because there was a negative shock.
But if you hire the same agent to do the same job year after year, and if the shock is serially uncorrelated between one year and the next, it gets a lot easier to solve the principal-agent problem. You tell the agent he has an annual production target p* for each year, and a cumulative target over T years of Tp*. He can miss the annual target, but if he misses the cumulative target by more than X% he will be fired. In the limit, as T goes to infinity, the average level of p must approach p*, for any finite X. And the agent knows that if he slacks off more this year he will have to work extra hard in future years to have the same probability of keeping the job.
This is probably one reason why people like to give repeated business to the same car mechanic.
Let p be the inflation rate, let s be the central bank's forecast error, and let m be the monetary policy instrument setting that will deliver p if s=0. Forecast errors should be mean-zero and serially uncorrelated if the central bank has rational expectations. Price level path targeting is like telling the central bank it will be fired if its cumulative misses of the inflation target p* add up to more than X%.
As Scott Sumner said a few months back (I can't find the link), level-path targeting keeps the central bank honest. It knows it will have to make up for past "mistakes" in future.