Learn Hardcore German: Nach dem Trocknen lege ich das Handtuch in den Schrank. - After drying, I put the towel in the cupboard.

nach is a preposition that always takes the dative case. The noun here is the nominalized infinitive das Trocknen (neuter). Dative singular of neuter is dem, so you get nach dem Trocknen. Other examples:

nach der Arbeit
nach dem Essen
nach dem Regen

Why is Trocknen capitalized?

Because it is a verb used as a noun (a nominalized infinitive). All nouns are capitalized in German. The other nouns here are Handtuch and Schrank.

Can I say Nach dem Trocken?

No. trocken is an adjective meaning dry. To express the process, use the nominalized verb das Trocknen. die Trocknung exists but sounds technical; in everyday language nach dem Trocknen is the natural choice. For extra clarity you can say nach dem Trocknen des Handtuchs.

Why in den Schrank and not im Schrank?

Because in is a two-way preposition. It takes:

accusative for movement toward a place (direction): Ich lege das Handtuch in den Schrank.
dative for location (no movement): Das Handtuch liegt im Schrank. (im = in dem)

Why in den Schrank and not ins Schrank?

ins is a contraction of in das, used only with neuter nouns. Schrank is masculine (der Schrank), so the correct accusative is in den Schrank. You could say ins Regal because Regal is neuter.

What case is das Handtuch here?

Accusative, as the direct object of legen. For neuter nouns, nominative and accusative look the same (das Handtuch), so there is no visible change compared to the dictionary form.

Why is the verb lege in second position after the opening phrase?

German main clauses follow the verb-second rule. The entire time phrase Nach dem Trocknen counts as one element in first position, so the finite verb lege must come next. The subject ich then follows: Nach dem Trocknen lege ich …. With neutral word order you can also say Ich lege ….

Can I place nach dem Trocknen somewhere else?

Yes. Common options include:

Ich lege nach dem Trocknen das Handtuch in den Schrank.
Ich lege das Handtuch nach dem Trocknen in den Schrank. Putting time information early often sounds natural in German. Within the middle field, the rough preference is Time–Manner–Place, but emphasis can change the order.

Why choose legen and not stellen or setzen?

legen: put something so that it lies flat or horizontally.
stellen: put something so that it stands upright.
setzen: set someone/something into a sitting position; more restricted for objects. A folded towel on a shelf is placed flat, so legen is the idiomatic verb. If you hang it up, you would use hängen.

Do I need a directional particle like hinein or rein?

Not needed. in den Schrank already expresses motion into something. hinein (standard) or rein (colloquial) can be added for emphasis or clarity: in den Schrank hinein.

How does Nach dem Trocknen compare to Nachdem es getrocknet ist?

nach + dative uses a noun phrase: compact and neutral in style (Nach dem Trocknen lege ich …).
nachdem introduces a subordinate clause: more explicit about the subject and typically uses the perfect to mark prior completion (Nachdem das Handtuch getrocknet ist, lege ich es in den Schrank.) Both are correct; choose based on style and need for explicitness.

Could I say Nach dem Abtrocknen?

Only if you mean the act of drying something by wiping it dry. abtrocknen usually involves an agent and a towel, e.g., Nach dem Abtrocknen der Gläser …. For a towel that has simply become dry, use Nach dem Trocknen.

Do I need a comma after the initial phrase?

No. A fronted prepositional phrase like Nach dem Trocknen is not set off by a comma. You would use a comma with a subordinate clause, e.g., with nachdem.

Any quick pronunciation pointers for this sentence?

ch in Nach is the ach-sound ; in ich it's the ich-sound [ç].
Final -en in legen is reduced; in Trocknen, many speakers make the second syllable very short or even syllabic.
Stress falls on the first syllables: Nach, Trocknen, Handtuch, Schrank.