Nach dem Essen möchte ich im Garten weiterzeichnen.

Because nach takes the dative case in this meaning.

The noun is das Essen in the nominative, but after nach it becomes dem Essen:

• nominative: das Essen
• dative: dem Essen

So nach dem Essen means after the meal / after eating.

Because Essen is being used as a noun, not as the verb essen.

German capitalizes all nouns. Here, das Essen means something like:

• the meal
• the eating

This is very common in German: a verb can become a noun. Compare:

• essen = to eat
• das Essen = the meal / eating

Here nach means after in a time expression.

So:

• nach dem Essen = after the meal / after eating

Be careful, because nach can also mean other things in other contexts, such as:

• to / toward: nach Berlin
• according to: nach dem Plan

In this sentence, it clearly has the time meaning after.

Im is a contraction of in dem:

• in dem Garten → im Garten

This is very common in German.

Here the phrase describes location: the speaker wants to do the drawing in the garden. Because it is a stationary location, German uses the dative:

• im Garten = in the garden

If it were movement into the garden, German would usually use the accusative:

• in den Garten = into the garden

So:

• Ich zeichne im Garten. = I draw in the garden.
• Ich gehe in den Garten. = I go into the garden.

Because German main clauses usually follow the verb-second rule.

That means the finite verb must come in the second position. In this sentence, the first element is:

• Nach dem Essen

So the finite verb must come next:

• Nach dem Essen möchte ich ...

Even though English often puts the subject first, German does not have to. If you start with a time phrase, the verb still stays second.

You could also say:

• Ich möchte nach dem Essen im Garten weiterzeichnen.

That version starts with the subject, but both are correct.

Because möchte is the finite verb, and the main action verb appears as an infinitive at the end.

German often does this with verbs like möchte, kann, will, muss, etc.

Structure:

• möchte = finite verb in second position
• weiterzeichnen = infinitive at the end

So:

• Nach dem Essen möchte ich im Garten weiterzeichnen.

This is similar to:

• Ich kann gut schwimmen.
• Sie will heute arbeiten.

Yes, in the infinitive here it is normally written as one word: weiterzeichnen.

It is built from:

• weiter = further / on / continue
• zeichnen = to draw

So weiterzeichnen means to continue drawing.

In a conjugated sentence without a modal-style verb, the weiter part often separates:

• Ich zeichne weiter. = I keep drawing / continue drawing.

But after möchte, the infinitive stays together at the end:

• Ich möchte weiterzeichnen.

Möchte is softer and more polite than will.

• ich möchte = I would like to
• ich will = I want to

For many learners, will looks like English will, but it does not mean future. It means want.

So:

• Ich möchte im Garten weiterzeichnen. = softer, more natural in many situations
• Ich will im Garten weiterzeichnen. = stronger, more direct

It suggests that the person has already been drawing and wants to continue after eating.

So the sentence does not just mean draw. It means:

• keep drawing
• continue drawing

That little piece weiter- adds the idea of continuation.

Yes. German word order is flexible, as long as the finite verb stays in second position in a main clause.

For example, these are also possible:

• Ich möchte nach dem Essen im Garten weiterzeichnen.
• Im Garten möchte ich nach dem Essen weiterzeichnen.

The meaning stays very similar, but the emphasis changes:

• Nach dem Essen ... emphasizes the time
• Im Garten ... emphasizes the location
• Ich ... is the most neutral starting point

Not most naturally in this sentence.

While Essen can sometimes refer to food, in nach dem Essen it usually means:

• after the meal
• after eating

So here it refers more to the event or mealtime than to the food itself.