Some sources of the hadith
The hadith can be found in other sources either in the same or similar wording or as parts of the hadith only. I'll try to stick on those who include the highlighted part: So you may find it in Sunan abi Dawod, Sunan ibn Majah,another version in Jami' at-Tirmdihi, -from here on only Arabic version available- Sunan ad-Darimi, Mosanaf ibn Abi Shabyah (Hadith #1), Mustadrak al-Hakim, and three quotes in Musnad Ahmad and four quotes in at-Tabaranis al-Kabir.
Off-topic but just an observation I found interesting while investigating
Note that at-Tirmdihi has narrated it once via abu Dawod and abu Dawud has compiled in his Sunan a version he has narrated via imam Ahmad all narrations are on the authority of Abu Razin Al-'Uqaili. Basically the two versions of imam at-Trimidhi show that he has one version with a high sanad (short narrator chain) and another with a longer chain in this is the case for all those who quoted different narrations most are narrated based on the narrator chain:
X -> Y -> Z -> from Ya'la ibn 'Ata -> from Waki' ibn 'Uds -> from his uncle -> from Abu Rain al-'Uqaili.
Were "Z" is either of Sho'abah شعبة or Hushaym هشيم. And "Y" might be abu Dawod or imam Ahmad in the shorter variation or again Hushaym who heard Sho'abah or other scholars as intermediates such as Hammad ibn Salamah.
The interpretation of the hadith
"And it is (as if it is) on the leg of a bird, as long as it is not spoken of. But when it is spoken of it drops."
Note that bird usually -you should rather think of a sparrow- has very small "legs" so that they are physically not able to carry big or heavy things. And if they try this these things may drop down. This might be a helpful input to understand the meaning, but let's check some hadith interpretations:
Al-Mubarakpuri عبد الرحمن المباركفوري the author of Tuhafat al-Ahwadhi تحفة الأحوذي -commentary on Jami' at-Tirmidhi- commented the hadith as follows (My own translation take it carefully):
( على رجل طائر ) هذا مثل في عدم تقرر الشيء أي لا تستقر الرؤيا قرارا كالشيء المعلق على رجل طائر ذكره ابن الملك ، فالمعنى أنها كالشيء المعلق برجل الطائر لا استقرار لها ، قال في النهاية : أي لا يستقر تأويلها حتى تعبر ، يريد أنها سريعة السقوط إذا عبرت ، كما أن الطير لا يستقر في أكثر أحواله فكيف يكون ما على رجله
(on the leg of a bird) this is an example on things that are not yet decided (or put in the place or position) as it means the dream is not decided (out in final position) the same as a thing that is appended at the leg of a bird this was said by ibn Malik -linguist-. So the meaning is that a dream has not in a study state as things which are appended at the leg of a bird are. And in an-Nihayah it was explained: This means the interpretation is not yet fixed (decided) unless it was made, so if it was interpreted it will fall down rapidly, also the bid is not settled in one position in most of his circumstances.
( ما لم يحدث ) أي ما لم يتكلم المؤمن أو الرائي ( بها ) أي بتلك الرؤيا أو تعبيرها
(as long as it is not spoken) (of) Means as long as the believer or the person who saw the dream has not spoken of it by either telling or interpreting it.
( فإذا تحدث بها سقطت ) أي تلك الرؤيا على الرائي يعني يلحقه حكمها ، وفي رواية أبي داود قال : الرؤيا على رجل طائر ما لم تعبر فإذا عبرت وقعت ، قلت : هذه الرواية تدل على أن المراد بقوله ما لم يحدث ما لم يتكلم بتعبيرها
(But when it is spoken of it drops) this means that the interpretation of the dream will drop on him. And in the narration of abu Dawood:
"The vision flutters over a man as long as it is not interpreted , but when it is interpreted, it settles."
This interpretation shows that what is meant as long as it is not spoken of refers to speaking out an interpretation (of the vision).
Note that this rather blurry explanation may lead to the conclusion that an interpretation the first who interprets a dream may change your predestination, so we need to further explanations!
One could find similar statements in 'Awn al-Ma'bud عون المعبود the commentary of Sunan abi Dawod by al-Aabadi but in Hashiyat as-Sindi on the Sunan of ibn Majah حاشية السندي على ابن ماجه the information one could deduce is less.
The conclusion one could make so far from this hadith is that the first interpretation of a dream falls on the person who had the vision as a rule so it is the most decisive therefore one should be careful in telling dreams, interpreting them or asking another person to interpret it. One should make a good choice the ahadith quotes: intelligent one (knowledgeable) or one who has judgement or wise one, a beloved one (not an enemy or somebody who might use it against you)
Therefore Mualla 'ali al-Qari quoted in his Mirqat al-Mafatih sharj Mishkat al-Masabyh مرقاة المفاتيح شرح مشكاة المصابيح the important advise of Ya'aqub to his son Yusuf:
... O my son, do not relate your vision to your brothers or they will contrive against you a plan. ... (12:5)
He also added an interpretation of at-Tibi saying that the example is one of a bird that is flying fast and has something on its leg which may fall easily by any simple movement also added an explanation the predestination or settling of the vision (My own translation atke it with the necessary precaution):
وهي أن الرؤيا مستقرة على ما يسوقه التقدير إليه من التعبير ، فإذا كانت في حكم الواقع قيض ، وألهم من يتكلم بتأويلها على ما قدر فيقع سريعا ، وإن لم يكن في حكمه لم يقدر لها من يعبرها .
Among the examples made is that the vision is settled on that what the predestination has lead the interpretation to, so if the interpretation was describing the real situation it was fixed, and the person who made this interpretation has been inspired (or given ilhaam) to what was predestined it will fall (be real) quickly, and if his verdict (interpretation) was not then the right person to interpret it was not predestined for it.
Therefore imam al-Bukhari quoted in his Sahih the hadith of abu Bakr's interpretation:
... The Prophet replied, "You are right in some of it and wrong in some." Abu Bakr said, "O Allah's Prophet! By Allah, you must tell me in what I was wrong." The Prophet (ﷺ) said, "Do not swear". (Sahih al-Bukhari)
Imam an-Nawawi in his commentary on Sahih Muslim concluded from this: "The vision is not settled for the first interpreter by default, but only if he was correct in his interpretation"!
Imam az-Zamakhshari said (Again my own translation take it carefully):
ليس المعنى أن كل من عبَرها وقعت على ما عَبَر ، ولكن إذا كان العابر الأول عالماً بشروط العبارة ، فاجتهدَ ، وأَدَّى شرائطها ، ووُفق للصواب ، فهي واقعةٌ على ما قال ، دونَ غيره
This doesn't mean that any dream interpretation made will happen as interpreted but only if the first person who interpreted it had knowledge of the conditions (and rules) of dream interpretations, and made an effort (ijtihad), and made it based on this rulings, and was lead to the correct interpretation, then it would fall/happen as he said, but not for any other.
Source of the former statements of the imams an-Nawawi and az-Zamakhsahri: fatwa -in Arabic- islamqa #117665.
Therefore many scholars considered this hadith as rejcted:
“Interpret them (dreams) in accordance with the names of the things you see, and understand what is hinted at, and the dream is for the first interpreter.” (Sunan ibn Majah)
A vision might be interpreted wrongly or correctly. The later can only be the case if we find a person that has enough knowledge of dream interpretation. And it doesn't change the predestination by any means, but telling dreams is risky therefore one should be aware that one must chose whom to tell and that the first interpretation -if correct- might come true.
And to use your wording in the question title: If the dream interpreter has been guided or inspired (given ilham) to the correct interpretation it will easily and quickly fall and come true as if it was loosely fastened at the leg of a bird.