Sobald der Regen aufhört, graben wir im Beet weiter.

Because sobald der Regen aufhört is a subordinate clause. In German, the conjugated verb usually goes to the end of a subordinate clause.

Also, aufhören is a separable verb:

• main clause: Der Regen hört auf.
• subordinate clause: ..., sobald der Regen aufhört.

So in a subordinate clause, the prefix auf- stays attached to the verb, giving aufhört.

German main clauses follow the verb-second rule. The conjugated verb must be in the second position.

Here, the whole subordinate clause Sobald der Regen aufhört takes the first position. That means the verb in the main clause must come next:

• Sobald der Regen aufhört, graben wir im Beet weiter.

So graben comes before wir. This is very common when a sentence starts with a time clause, condition clause, etc.

German often uses the present tense to talk about the future when the time is clear from context.

Here, sobald already shows that the action will happen later:

• Sobald der Regen aufhört, graben wir im Beet weiter.

English often does something similar:

• As soon as the rain stops, we’ll continue digging.

German could use werden, but it usually sounds less natural here:

• ..., werden wir im Beet weitergraben.
  This is possible, but the simple present is more common.

Sobald means as soon as or once. It emphasizes that one action happens immediately after another.

So:

• Sobald der Regen aufhört ... = As soon as the rain stops ...

It is not quite the same as wenn:

• wenn can mean if, when, or whenever
• sobald is more specific and usually means as soon as

So sobald gives a stronger sense of immediate sequence.

Regen is a masculine noun: der Regen.

In this sentence, der Regen is the subject of the clause:

• der Regen aufhört = the rain stops

Since it is the subject, it is in the nominative case, and the masculine nominative article is der.

Im Beet means in the flowerbed / in the garden bed.

It is a contraction of:

• in dem Beet → im Beet

Here, in is used for location, not movement, so it takes the dative:

• dem Beet

A Beet is a planted area in a garden, such as a flowerbed or vegetable bed.

Because this sentence describes where they are digging, not movement into the bed.

German often distinguishes:

• wo? = where? → usually dative
• wohin? = to where? → usually accusative

So:

• im Beet = in the bed / in the flowerbed → location
• ins Beet = into the bed → movement into it

Here they are already working there, so im Beet is correct.

Weiter here means further, on, or continue.

So weitergraben or graben ... weiter means:

• to keep digging
• to continue digging

In this sentence:

• graben wir im Beet weiter = we continue digging in the flowerbed

It shows that the digging had already started before the rain.

Yes, that is also possible.

Both are grammatical:

• ..., graben wir im Beet weiter.
• ..., graben wir weiter im Beet.

The difference is mainly about focus and rhythm:

• im Beet weiter can put a little more attention on where
• weiter im Beet can put a little more attention on continuing

In everyday German, both word orders are natural.

Yes. In German, a subordinate clause is separated from the main clause by a comma.

So this is correct:

• Sobald der Regen aufhört, graben wir im Beet weiter.

That comma is required.

Because graben can be used without a direct object when the meaning is clear.

Here it means something like:

• to dig
• to keep digging

The location im Beet already tells you where the digging happens, so no object is necessary.

You could add one in other contexts:

• Wir graben ein Loch. = We are digging a hole.

But in this sentence, German naturally leaves the object out.