TSP is actually very amenable to heuristics and state of the art branch-and-bound algorithms can often find optimal solutions even for instances with thousands of points.
With that many points, how do you prove that your solution is optimal?
Often in approximations you can prove an upper bound on the optimal solution, even if you don't know the optimal solution itself.
An easy example is by using duality when solving linear programs.
And sometimes your upper bound is close enough to what you already have so you can just say something like "well my solution gets 190 points, I have an upper bound of 190.5 points, and all point rewards are integral so I must have the optimal solution."
TSP (in the traditional sense of finding an optimal solution) is NP-complete, so (by definition of NP-complete) it's much easier to check a proposed correct answer than to find that answer.
This is easy to understand with integer factorization: It is very hard to factor large integers. Yet, if I where to give you two numbers, it is very easy to verify that their product is the integer you where looking at. So factorization is easy to verify, but hard to do (side-note: This is an intentional illustrative simplification. Integer factorization is not known to actually be "hard" and it is not known whether "hard" is even a thing. See below).
And that's the definition of an NP problem: It has a polynomial time deterministic algorithm to verify a solution.
An NP-complete problem is a problem that is at least as hard as any NP problem. That is, if you have an algorithm for an NP-complete problem, you can take any NP problem, transform it efficiently into that problem, use your algorithm and transform the solution efficiently back. Thus, if you've solved an NP-complete problem efficiently, you can solve all NP problems efficiently. TSP is NP-complete. Whether or not Integer Factorization is NP-complete is unknown.
Lastly, there is the question of whether there are problems that are in NP (that is, have a polynomial algorithm to verify a solution) but not in P (that is, have a polynomial algorithm to find a solution). That's the famous P vs. NP question, which is currently undecided.
With that many points, how do you prove that your solution is optimal?