Tôi gấp áo rồi cất vào ngăn kéo.

It can mean both, depending on context.

In gấp áo, gấp is a verb meaning to fold. Since áo is a piece of clothing, the meaning is clearly fold clothes / fold a shirt.

In other contexts, gấp can be an adjective meaning urgent, as in việc gấp = urgent work.

So here, the grammar and the object after it tell you that gấp means fold.

Literally, áo usually means shirt, top, or garment.

So the most literal reading of gấp áo is fold a shirt / fold the shirt.

If you want to say clothes more generally, Vietnamese often uses quần áo. For example:

• Tôi gấp quần áo rồi cất vào ngăn kéo. = I fold the clothes and put them in the drawer.

So if your translation says clothes, that may be a looser translation, but the word in the sentence is specifically áo.

Here rồi means something like then, and then, or after that.

It links the two actions in sequence:

1. Tôi gấp áo = I fold the shirt
2. rồi cất vào ngăn kéo = then put it into the drawer

So rồi shows that the second action happens after the first one.

Important: this is not exactly the same as sentence-final rồi, which often means already or marks completion. In this sentence, rồi is mainly connecting two actions.

Because the subject stays the same.

In English, we usually say I fold the shirt and then I put it in the drawer, but in Vietnamese, once the subject is clear, it often does not need to be repeated.

So:

• Tôi gấp áo rồi cất vào ngăn kéo.

is understood as:

• Tôi gấp áo rồi tôi cất vào ngăn kéo.

The second tôi is omitted because it would sound repetitive.

Because the object is already understood from the first clause.

The full version could be:

• Tôi gấp áo rồi cất áo vào ngăn kéo.

But repeating áo is often unnecessary if the meaning is obvious. Vietnamese very commonly drops repeated objects when they are easy to understand from context.

So cất vào ngăn kéo means put it away into the drawer, with áo understood.

Here cất means put away, store, or put somewhere neatly/out of the way.

So cất vào ngăn kéo suggests putting the shirt away in the drawer, not just placing it there casually.

Compare:

• để = put / leave
• cất = put away / store

For example:

• để trên bàn = put/leave it on the table
• cất vào ngăn kéo = put it away in the drawer

So cất gives a stronger sense of tidying or storing.

Because vào means into.

The phrase vào ngăn kéo means into the drawer. It shows the direction or destination of the action.

So:

• cất vào ngăn kéo = put away into the drawer

Without vào, the sentence would sound less complete in this context, because cất often works naturally with a phrase showing where something is being put.

Ngăn kéo is the normal Vietnamese word for drawer.

You can think of it as a fixed vocabulary item. Even though the parts may originally suggest something like a pull-out compartment, learners should usually just remember ngăn kéo as the standard word for drawer.

So:

• vào ngăn kéo = into the drawer

Vietnamese normally does not use articles like a, an, or the.

So áo can mean:

• a shirt
• the shirt
• sometimes just shirt/clothing

The exact meaning depends on context.

That is why Tôi gấp áo rồi cất vào ngăn kéo can be translated naturally into English in more than one way, such as:

• I fold the shirt and put it in the drawer.
• I fold a shirt and put it in the drawer.

English must choose an article, but Vietnamese often does not.

Not in this sentence.

After verbs, Vietnamese often uses a bare noun directly:

• gấp áo
• mặc áo
• giặt áo

That is completely natural.

If you want to make the noun more specific, you can use a classifier:

• gấp cái áo = fold the shirt
• gấp một cái áo = fold a shirt

So cái is possible, but it is not required here.

Yes. A few natural alternatives are:

• Tôi gấp áo rồi cất áo vào ngăn kéo.
  This repeats the object for extra clarity.

• Tôi gấp cái áo rồi cất vào ngăn kéo.
  This makes áo sound more like a specific shirt.

• Tôi gấp quần áo rồi cất vào ngăn kéo.
  This changes the meaning to clothes in general.

The original sentence is already natural. These are just slightly different ways to express related meanings.