bǎ xiánzhì wánjù dǎbāo yǐhòu, wǒmen téngchū le hěnduō kōngjiān.

把 is used to highlight what is being handled and what happens to it.

The pattern is:

把 + object + verb/result

So here:

把 + 闲置玩具 + 打包

means take the unused toys and pack them up.

This structure is very common when:

• the object is specific and known
• the action changes, moves, disposes of, or affects that object
• there is often a clear result

Using 把 makes the sentence feel more like we dealt with the unused toys by packing them up.

闲置 means left unused, idle, or not being used for the moment.

So 闲置玩具 means:

• unused toys
• toys sitting around not being used

It sounds a bit more formal or written than just saying the toys are not being played with. You often see 闲置 with things like:

• 闲置房屋 — unused house/property
• 闲置设备 — idle equipment
• 闲置物品 — unused items

Yes, 打包 means pack up or bundle up.

In this sentence, it suggests gathering the toys together and putting them into boxes, bags, or packages so they are stored away.

Depending on context, 打包 can also mean:

• to pack things up
• to package something
• at a restaurant, to pack leftovers to take away

Here it clearly means pack up the unused toys.

以后 means after or afterwards, depending on where it appears.

Here it comes after a verb phrase:

把闲置玩具打包以后

This means:

after packing up the unused toys

So the structure is:

[action] + 以后, [main clause]

Examples:

• 吃完饭以后，我们去散步。 — After eating, we went for a walk.
• 下课以后，我去图书馆。 — After class, I go/went to the library.

In your sentence, 以后 connects the first action with what happened next.

Yes, in this sentence 以后 and 之后 are both natural.

So you could say:

把闲置玩具打包之后，我们腾出了很多空间。

That also means After packing up the unused toys, we freed up a lot of space.

In many everyday sentences, 以后 and 之后 are very close in meaning.
A rough tendency is:

• 以后 is extremely common in speech
• 之后 can sound a bit more formal or written

But both work here.

腾出 means clear out so as to make available, free up, or make room.

So:

腾出了很多空间

means:

freed up a lot of space

This is more specific than 有了很多空间.

Compare:

• 有了很多空间 — now there is a lot of space
• 腾出了很多空间 — we created that space by moving or removing things

So 腾出 emphasizes the result of clearing something away.

The 了 marks the action as completed and presents the result as having occurred.

So:

我们腾出了很多空间

means:

• we freed up a lot of space
• the action was completed
• the result now exists

Without 了, the sentence would sound more like a general statement, habitual action, or something less tied to a completed event.

Because this sentence describes what happened after packing up the toys, 了 fits very naturally.

Because 空间 is often treated as something that can be described directly with quantity words like 很多.

So:

很多空间 = a lot of space

You do not need a classifier here.

This is similar to:

• 很多时间 — a lot of time
• 很多机会 — many opportunities
• 很多钱 — a lot of money

Even though English uses a lot of, Mandarin does not always need a separate measure word in these cases.

They are related, but not exactly the same.

空间 means:

• space
• room
• physical capacity or open area

地方 usually means:

• place
• spot
• area/location

In this sentence, 空间 is better because the idea is that packing away toys created more usable room.

So:

• 腾出了很多空间 — freed up a lot of space
• 腾出了很多地方 — possible in some contexts, but less natural here

Use 空间 when talking about room, capacity, or openness.

This is normal in Chinese.

In the first part:

把闲置玩具打包以后

the subject is understood from context. Then the main clause gives the explicit subject:

我们腾出了很多空间。

Chinese often omits information that is easy to understand.
So the full logical meaning is:

After we packed up the unused toys, we freed up a lot of space.

But repeating 我们 in the first part is usually unnecessary.

Yes. You could say:

打包闲置玩具以后，我们腾出了很多空间。

This is understandable and grammatical.

However, 把闲置玩具打包以后 sounds a bit more natural if you want to emphasize the handling of those specific toys. The 把 version makes the object more prominent and gives a stronger take the toys and do something to them feeling.

So both are possible, but the 把 version is especially good here.

The overall pattern is:

[After doing action A], [subject] + [completed action B + result]

More specifically:

把 + object + verb + 以后，subject + verb + 了 + result/object

In this sentence:

• 把闲置玩具打包以后 — after packing up the unused toys
• 我们腾出了很多空间 — we freed up a lot of space

This is a very common way to describe:

1. an earlier action
2. a later result

So the sentence flows as:

After doing X, we achieved Y.