When the Constitution is amended, it changes. That's it. Because the 13th Amendment is an Amendment to the Constitution, it changes the document. That is why there is a provision to amend it in the original, so that these things can be written into the Constitution. The 13th and all other Amendments are Constitutional because they are part of the Constitution, and therefore legal.
The 10th Amendment is not the supreme Amendment, and it does not prevent any other Amendments from being written in. It does prevent the Federal government from usurping state powers except as otherwise Constitutionally given to the Federal government. However, when an Amendment is passed and goes through the whole process, what the Amendment regulates is now written into the Constitution. If it gives more power to the Feds, then the 10th does not apply because those powers are now Constitutionally given to the Federal Government.