Explanation based on the related verse of the qur'an
The hadith was compiled in Sahih al-Bukhari in the chapter about tafeer and the verse which is referred to here is:
You are the best nation produced [as an example] for mankind. You enjoin what is right and forbid what is wrong and believe in Allah . If only the People of the Scripture had believed, it would have been better for them. Among them are believers, but most of them are defiantly disobedient. (3:110)
This verse describes the Muslims as the best nation (Ummah) produced for mankind: (as long as) they enjoin right and forbid what is wrong and believe in Allah. It also shows that if the other nations especially the people of the book believed -in the same manner- it would have been better for them. But among them -not all of them- there are defiantly disobedient people.
And if you read the next verses you will see how these disobedient people fight and act against Muslims until the Muslims will be victorious over them and humiliate them -in first place-. This already might explain the reference and use of the statement:
"... chains on their necks …"
to some extent.
As if we think further and spin out the thought: what will happen to those who were humiliated after a war? Wouldn't they either go back home humiliated and defeated or stay at the battlefield and become war prisoners (get enchained)? Many of the war prisoners at the time of the prophet () later accepted Islam, while they were literally chained after the battle!
Also humiliation refers to the use of force as you may do something against your wishes and be humiliated this way!
Explanation based on a similar hadith
As ibn Hajar didn't care to explain the specific part of the statement you are asking about we need to look further:
In another hadith you may read:
"Allah wonders at those people who will enter Paradise in chains." (Sahih al-Bukhari)
According to ibn Hajar al-'Asqalani in his fath al-Bary فتح الباري شرح صحيح البخاري (see here in Arabic) after explaining that the described situation is a reference to their situation in this life (not in the hereafter).
He quoted first a statement of ibn al-Jawzi ابن الجوزي saying: that those people have fought Muslims and ended up as prisoners of war and lived and witnessed the Muslims, then believed and reverted to Islam.
This was one explanation while at-Tyybi الطيبي said it could be a reference to the act of "pulling" which often is necessary to help people either avoid doing bad deeds or to help them doing good deeds (among them doing kufr or converting to Islam).
He later said that as the hadith -we are discussing- was quoted in the book of tafseer it must be taken literally quoted a hadith which seems to appear in Munsad al-Bazzar which say that some people of our prophets () ummah will become Muslims against their whishes (by force).
While Ibrahim al-Harbi إبراهيم الحربي held the opinion that the enchainment is not to be taken literal as it refers to guidance to Islam by force.
Finally he quoted opposite explanation held by others that says that it refers to Muslim war prisoners who either die as war prisoners of non-Muslims or will be killed in the described state (with chains on their necks) and therefore will enter Jannah in this state.