bonyarisite ite, ie ni kagi wo kakeruno wo wasuremasita.

It means something like being absent-minded, not paying full attention, or spacing out.

In this sentence, ぼんやりしていて sets up the reason or background for what happened next:

• ぼんやりしていて、家に鍵をかけるのを忘れました。
• I was absent-minded, and I forgot to lock the house.

So ぼんやりしていて is not the main action. It describes the state the speaker was in when they forgot.

Both come from する.

• して = the て-form
• していて = している in て-form

Here, ぼんやりしている means to be absent-minded / to be in a dazed state.
When it becomes ぼんやりしていて, it connects that ongoing state to the next clause.

So the nuance is:

• ぼんやりして = simply connecting actions/states
• ぼんやりしていて = emphasizing that the speaker was in that state

In other words, していて sounds more like:

• while being absent-minded
• because I was absent-minded

ぼんやり is an adverb-like or mimetic expression often used to describe a vague, unfocused, absent-minded state.

You will often see it with する:

• ぼんやりする = to daydream, to be absent-minded, to zone out

Examples:

• ぼんやりしています。 = I’m spacing out.
• 窓の外をぼんやり見ていた。 = I was gazing vaguely out the window.

So in this sentence:

• ぼんやりしていて = being spaced out / absent-minded

In Japanese, 鍵をかける means to lock or more literally to put/apply a lock.

The place or target being locked is often marked with に:

• ドアに鍵をかける = lock the door
• 家に鍵をかける = lock the house

So:

• 家 = house
• に = marks the target/location
• 鍵をかける = apply the lock

This may feel different from English, where we directly say lock the house.
Japanese often phrases it more like put a lock on the house.

Literally, it is something like to put/apply a lock.

Breaking it down:

• 鍵 = key / lock
• を = object marker
• かける = to hang, apply, put on

In this expression, 鍵をかける is a set phrase meaning:

• to lock
• to lock up

Even though 鍵 can mean key, in this expression the whole phrase means the act of locking.

This is a very common pattern:

• dictionary form + のを忘れる
• to forget to do ...

So:

• 鍵をかける = to lock
• 鍵をかけるの = the act of locking
• 鍵をかけるのを忘れました = forgot the act of locking = forgot to lock

The の turns the verb phrase into something noun-like, so it can be the object of 忘れる.

You can think of it like:

• I forgot [locking the house].

This grammar is very common:

• 宿題をするのを忘れた。 = I forgot to do my homework.
• 電話するのを忘れました。 = I forgot to call.

Yes, you can also say:

• 家に鍵をかけることを忘れました。

This is grammatically correct and means the same thing.

However, の often sounds a bit more natural and conversational in cases like this, especially with everyday actions.
So:

• Vるのを忘れる = very common in speech
• Vることを忘れる = also correct, sometimes slightly more formal or abstract

In this sentence, の is a very natural choice.

Yes, the subject is omitted.

Japanese often leaves out the subject when it is clear from context.
Here, the sentence naturally implies I:

• (私は) ぼんやりしていて、家に鍵をかけるのを忘れました。

In English, you usually need to say I forgot, but in Japanese, 私は is often unnecessary if the speaker is obviously talking about themself.

Because the speaker is talking about a completed event: they forgot.

• 忘れます = forget / will forget
• 忘れました = forgot

The sentence describes something that already happened:

• The speaker was absent-minded
• As a result, they forgot to lock the house

So the past tense is the natural choice.

Also, ました makes it polite:

• 忘れた = plain casual
• 忘れました = polite

Here it strongly suggests a reason or cause.

So the sentence is understood as:

• Because I was absent-minded, I forgot to lock the house.

The て-form can connect clauses in several ways, including:

• sequence
• accompanying circumstance
• reason/cause

In this sentence, the most natural reading is causal:

• ぼんやりしていて → because I was spacing out

Usually, 家に鍵をかける is the more natural expression for locking the house.

Be careful with 家の鍵:

• 家の鍵 usually means the house key

So:

• 家に鍵をかける = lock the house
• 家の鍵 = the key to the house

If you say 家の鍵をかける, it may sound odd or confusing, because it can seem like you are doing something to the key itself, not locking the house.

The sentence breaks down like this:

• ぼんやりしていて、
  = being absent-minded / because I was absent-minded

• 家に鍵をかける
  = lock the house

• のを
  = turns 家に鍵をかける into the object of forget

• 忘れました
  = forgot

So the full structure is:

• [Reason/background] + [action nominalized] + を忘れました

Or more simply:

• Because I was absent-minded, I forgot to lock the house.