Since magnetic field lines are the path taken by a hypothetical North Pole when it is in range of a magnetic field of a magnet, it is clear that the direction of hypothetical North Pole would be from North Pole to South Pole of a magnet not even outside the magnet but inside the magnet too, because North Pole of magnet will repel the hypothetical North Pole inside and outside the magnet and would be attracted by the South Pole of the magnet.
In nature, there is no magnetic monopole discovered yet. All of the magnets we have are created by certain kind of current (like the spin of an electron). Thus, the prototype of a magnet is a solenoid.
Now, there are a bunch of ways to argue the direction of the magnetic field in the solenoid. If you took introductory physics before, please use Biot-Savart. If not, the most intuitive way is probably assuming that magnetic field should be smooth (this is based on the assumption of no magnetic monopole). The north pole is defined as where the magnetic field comes out. Due to the smoothness assumption, even when you go into the solenoid a little bit (from north pole), the direction of the magnetic field should be the same, which is now "pointing toward north pole". Thus, it should be pointing from south pole to north pole inside.
Take a large number of small magnets, all pointing in the same direction, and glue them together. You get something like:
This is a pretty good model for a typical bar magnet. The individual small magnets are individual atoms. Now place your hypothetical north pole inside this larger magnet, and you can see which way it is being pushed.
So far I quoted a picture also mentioned in the answer of HolgerFiedler. But I would like to take this a little further.
The next question might be, 'but what about those little magnets, what is inside them?' The answer to this question is that the magnetism of individual atoms is caused partly by intrinsic magnetism which particles such as electrons have, and partly by currents in the atom, associated with the charge on the electron and its motion. Both of these require a quantum physical description, but both respect the equations of classical electromagnetism called Maxwell's equations, and ultimately this lets you know that the lines of magnetic field $\bf B$ will always form continuous loops, never come to stop or a start. In the case of a single electron, the loops run right down to the electron itself. You can imagine the electron as spread out a little owing to its wavefunction being not perfectly point-like, and then the magnetism of the electron is also like the picture given above; that is, it is like a continuously spread-out bunch of tiny magnets all side by side. When the electron also moves, such as in the current loops found inside atoms, this adds a further contribution which you can model as like a solenoid.
Far away from any scientific explanation I want to give one easy to understand answer. Every magnet is a body and has an "inside" and an "outside". Two magnets influencing each over through their outside regions.
First about the "outside". If you were the first why find out that
- any magnet has an axis which will be aligned by another magnet and
- one end of the axis will be attracted to the other magnet while the over end will be strongest repealed
you would have the choose and the right to call the two poles how you want: North and South, Rede and Black, .... And you have the obligation to find a method for all other interested people to identify the poles of their magnets in the same manner you name the poles. For magnets this is easy because the earth has its own magnetic field and for a compass needle it is easy to define a north and a south pole.
Why I tell you this? Because sometimes teachers try to say you that poles not exist at all and this is justifiable for some understanding of magnets.
Now about the "inside". A permanent magnet is made of atoms with aligned electrons and these aligned electrons are the reason that a body is a permanent magnet. Electrons have the intrinsic (existing independent from external influences) property of a magnetic dipole moment, they are tiny magnets. In permanent magnets the alignment of the involved electrons is a self holding process.
Now imagine you have a lot of bar magnets. You stick them together and what you get is a good model of what happens inside any permanent magnet.
BTW, magnets have a saturation and if you search for images where iron shavings show the field lines you will see that they start and end not only on the plane ends of the bar magnet. North and south pole are directions and not unambiguously in their extents.