$P$ and $NP$ are classes of decision problems. The result of an algorithm for a decision problem is either "YES" or "NO". Even for a problem in $P$, such an answer cannot lead to a quick verification.
An instance of the decision problem version of TSP is "Given a collection of cities and intercity distances, is there a tour with total length less than $k$?", where $k$ is a constant specified in the instance. The result is "YES" or "NO". In neither case does the answer lead to a quick verification of the correctness of the answer.
The promise that you ask about is this: Given a particular proposed tour, one can in polynomial time:
- Determine that the proposed tour actually is a tour -- visits all the cities and only traverses intercity routes that exist (sometimes "that have finite distances" when one encodes missing routes as having length $\infty$).
- If so, determine that the length of the route is shorter than the constant $k$ in the problem instance.
Neither an answer of "YES" or "NO" provides a proposed tour.
The value of the model of $NP$ that you are using is that it encodes a way to make a solver: for each possible tour (typically an exponentially large set to iterate over) check to see if it is a tour and if its length is $< k$. If so, report "YES". If we exhaust the collection of possible tours without reporting "YES", report "NO".
Note that this model suggests that the the difficulty in fast solution is not that checking the conditions takes a lot of time. The difficulty in fast solution is that there are too many potential tours to search through. So, if we could find some really, really smart way to restrict our search to only a tiny subset the collection of potential tours, we would have a fast solution for an $NP$ problem.
Binary search in a sorted list is an example where one has a smart way to search through the list evaluating only logarithmically many (in the length of the list) comparisons rather than linearly many comparisons. From this point of view, the TSP problem is hard because we don't know a substantially faster way to search through the proposed tours of every possible TSP problem instance.