Learn German: Der Lehrer steht vorne. - The teacher is standing at the front.

Because the teacher isn't moving! The words hin (away from you) and her (towards you) are used for direction and movement. The word vorne (at the front) describes a fixed position. Because the teacher is just standing there, we use a positional word instead of a directional one.

How would I change this sentence to say the teacher is walking to the front?

You would need to swap the position word (vorne) for a directional phrase, and change the verb to one of motion. You would say Der Lehrer geht nach vorne (The teacher is walking to the front).

I know vor means "in front of". What is the difference between vor and vorne?

vor is a preposition, meaning it always needs a noun attached to it to make sense (for example, vor der Klasse means "in front of the class"). vorne is an adverb of location that stands entirely on its own to mean "at the front" or "in the front".

The English translation says "is standing", but I only see steht. Doesn't that just mean "stands"?

It means both! German doesn't have an exact equivalent to the English "-ing" form (the continuous tense). The single verb steht translates to both "stands" and "is standing", depending on what sounds best in English.