Learn Swedish: Han var sjuk igår, så han slipper klippa gräset. - He was sick yesterday, so he does not have to cut the grass.

Because slipper is a modal-like verb. Just like kan (can) or måste (must), verbs like slipper (not have to) and brukar (usually do) take the bare infinitive. You put the next verb, klippa, directly after it without an att.

What exactly does slipper mean? Is it related to slipping?

It is a false friend! The verb slippa means to not have to do something, to be spared from it, or to get out of an unpleasant task. In this sentence, it means he is spared from the chore of cutting the grass.

Why does the sentence use gräset instead of just gräs?

Gräs (grass) is an ett-word. Because we are talking about a specific lawn, we need the definite form. You add the suffix -et to make it gräset (the grass).

After the comma and så, why is the word order han slipper instead of the verb coming first?

The word så (so) is a coordinating conjunction that connects two independent main clauses. Because it joins two complete sentences, the second clause starts fresh with normal subject-verb order, which is why the subject han comes right before the verb slipper.