Learn German: Es tut mir leid. — Das macht nichts. - I am sorry.

In German, you don't literally say "I am sorry." Instead, the phrase is Es tut mir leid, which translates roughly to "It does me sorrow." It is a fixed expression you just have to memorize!

What is the word mir doing in the first sentence?

mir is the form of "I" that means "to me" or "for me" (called the dative case). Because the literal meaning of the phrase is "It does sorrow to me," you must use mir instead of ich (I).

How does Das macht nichts mean "It doesn't matter"?

Literally, Das macht nichts means "That makes nothing." In German, this is the standard, everyday way to say "Never mind," "No worries," or "It doesn't matter" when someone apologizes.

Why do we use tut (from tun) instead of macht (from machen) in the first part?

Both tun and machen can translate to "to do" or "to make" in English, but they are used in different set phrases. For feeling sorry, German always pairs leid (sorrow) with tun, never machen.