Learn Hardcore Hebrew: השיעור מתחיל בעוד חמש דקות.

שיעור usually means lesson, class, or sometimes lecture period.
With the definite article ה־, השיעור means the lesson / the class.

In this sentence, the most natural English meaning is the class or the lesson.

What does the ה־ at the beginning do?

ה־ is the Hebrew definite article, meaning the.

So:

שיעור = a lesson / a class
השיעור = the lesson / the class

It is attached directly to the noun, instead of being a separate word like English the.

Why is it מתחיל and not מתחילה?

Because שיעור is a masculine singular noun.

The verb form מתחיל agrees with the subject:

שיעור = masculine singular
therefore מתחיל = masculine singular

If the subject were feminine singular, you would use מתחילה.

What does מתחיל mean here?