Let's use Traveling Salesman as the example, unless you think there's a simpler, more understable example.

My understanding of P=NP question is that, given the optimal solution of a difficult problem, it's easy to check the answer, but very difficult to find the solution.

With the Traveling Salesman, given the shortest route, it's just as hard to determine it's the shortest route, because you have to calculate every route to ensure that solution is optimal.

That doesn't make sense. So what am I missing? I imagine lots of other people encounter a similar error in their understanding as they learn about this.

Your version of the TSP is actually NP-hard, exactly for the reasons you state. It is hard to check that it is the correct solution. The version of the TSP that is NP-complete is the decision version of the problem (quoting Wikipedia):


  The decision version of the TSP (where given a length L, the task is to decide whether the graph has a tour of at most L) belongs to the class of NP-complete problems.


In other words, instead of asking "What is the shortest possible route through the TSP graph?", we're asking "Is there a route through the TSP graph that fits within my budget?".
