They were very modest in appearance, slim middleedium-sized birds.

The birds' name turned out to be 'mourning dove'.

They saying hoot, hoot, hoot, like owls, but in a higher tone, then once in the phrase it becomes about one octave higher and they saying it like 'He'.

The original is fine; here is one more option: Sounds that I Hear 2 [A 2nd revision]

This has already become decades ago by now, but around that time there was aCLEARER: For some decades now, pairs of birds cominghave come almost every day and recitto my my kitchen window, chirping an interesting song by the kitchen windows.

COMMENT: I am guessing than *more than one* pair of birds came over the years. Also, the plural "interesting songs" is better than "an interesting song" since they can sing *more than one* song.

Once when I stepped out onto the small balcony in the middle of themy apartment building, I had a chance to see them atfrom a very short distance.

They were sitting on thea fence there show, displaying their backs to me, and I saw their long tails.

Several seconds passed before they flew away, saying something like "ka-ka-ka-ka-ka-ka...."

NOTE: The expression "something like" is a good animal noises.

The name of the birdsSMOOTHER: They turned out to be 'mourning dove's.

NOTE: You can delete the quotation markers from this sentence.

The birds' name turned out to be 'mourning dove'.[DELETE: This information has been mentioned already.]

It was a very unexpected name toThat name surprised me because they siang with soft voices but in a, slightly humorous mannerways.

They say hoot, hoot, hoot, like owls, but in a higher tone, then once in the phrase it becomes about one octave higher andlike they say likeound 'Hhe'.

IDEA: It might be good to write the IPA pronunciation since the word "he" can be pronounced several different ways.

ThisIt has already becomen decades ago by now, but around that time, there was a pair of birds comingthat would come almost every day and recitinge an interesting song by the kitchen window.

They saygo hoot, hoot, hoot, like owls, but in a higher tone, then once in the phrase it becomes about one octave higher and they sayit sounds like 'He'.

This hwas already become decades ago by now, but around that time, there was a pair of birds comingwould come almost every day and recitinge an interesting song by the kitchen window.

The phrase "This has already become decades ago by now" is grammatically correct, but perhaps not the most natural choice.
"This was already decades ago now" has a more recollective, storytelling nuance perhaps.
I changed "coming" and "reciting" to "would come + recite" as this tense reinforces a repeated or habitual action in the past.

They were sitting (or perched) on the fence there showing their backs to me, and I saw their long tails.

"Perhced" is commonly used to describe stationary birds, but "sat" is also fine. "Sit perched" is also used!

Several seconds passed before they flew away saying/singing/with a kakakakakaka....

[Verb] with a [onomatopoeia] is somewhat common expression. Like "going out with a bang."

It was a very unexpected name to me because they sing with soft voicessoftly but in a humorous manner.

It's not common to use "voice," "said," or other words like these for animals, I think. At least not in this context.

They saygo hoot, hoot, hoot, like owls, but in a higher tone, then once in the phrase it becomes about one octave higher and they say likego 'He'.

Using "go" as an alternative for said to avoid using "say" for animals. If you want to personify them more, though, said is probably fine.

This has already become decades ago by now, but around that time there wasIt's been a long time now, but back then a pair of birds comingused to come by almost every day and recitwould sing an interesting song by the kitchen window.

They were slim middle-sized birds, very modest in appearance, slim middle-sized birds.

Once when I stepped out onto the small balcony in the middle of the apartment building, I had a chance to see them at a very short distancup close.

They were sitting on the fence withere showing their backs tourned toward me, and I saw their long tails.

They might have been inMaybe they were deep in thought.

It was a very unexpected name to me because they sing with soft voices but in a humorous mannerfunny way.

They say hoot, hoot, hoot, like owls, but in a higher tone, then once inbut after a few times the phrase it becomess raised about one octave higher and they sayand it sounds like 'He'.

This has already becomeIt was decades ago by now, but around that time there wasthere used to be a pair of birds comingthat came almost every day andto my kitchen window reciting an interesting song by the kitchen window.

The birds' name turned out to be 'mourning dove'.

either is fine, or "it turns out the birds are called "mourning doves"

It was a very unexpected name to me because they sing with soft voices but in a humorous manner.

the name was unexpected, judging from their soft but unusual singing?

They say hoot, hoot, hoot, like owls, but in a higher tone, then once in the phrase it becomes about one octave higher and they sayit sounds like 'He'.

I don't know who or what they were mourning about, but they have non't visited me for quite some time since.

ThisIt has already becomen decades ago by now, but around that time, there was a paircouple of birds coming almost every day and recitsinging an interesting song by the kitchen window.

Once, when I stepped out onto the small balcony in the middle of the apartment building, I hadgot a chance to see them at a very shortclose distance.

They were sitting on the *fence there showing their backs to me, and I saw; I got to see their long tails.

*railing

The name of the birds turned out to be 'mourning dove's.

IFor a bird's name, that was a vrathery unexpected name to me because they sing with soft voices but in athat sounds humorous manner.

They say hoot, hoot, hoot,hoot like owls, but inat a higher tone*, then once in the phrase it becomes about one octave higher and they say like 'He'a series of hoots, it becomes like a "He" but one octave higher.

*pitch?