On the authenticity of the narration
You should know that strictly speaking a narration is only a hadith (meaning a statement or act made by our prophet ()) when it is (correctly) attributed to the prophet () anything else is rather known as athar or khabar.
As for the case of the narration you quoted its attribution to the prophet is considered as weak by Sunni scholars and it is a mawquf hadith (which means in the terminology of hadith scholars: a statement of a sahabi) on the authority of 'Aishah () and her sister Asma' () and some attributed it to 'Umar ibn al-Khattab ().
Al hafidh al-'Iraqi quoted in his foot note on the hadith in imam al-Ghazaly's al-Ihya' the following comment of imam al-Bayhaqi:
قال البيهقي: وروي ذلك مرفوعاً، والموقوف أصح.
In the following if I quoted a translation from Arabic language then it is of my own take it with the necessary care!
It was narrated as marfo' (meaning attributed to the prophet()), but the mawquf version is more correct.
On the meaning and interpretation of the statement
Nevertheless there's actually a hadith with similar wordings which is a part of the prophet's so called last sermon (khutbat al-Wada') which also includes an explanation of the therein used expression:
“And indeed I order you to be good to the women, for they are but captives with you over whom you have no power than that, except if they come with manifest Fahishah (evil behavior). If they do that, then abandon their beds and beat them with a beating that is not harmful. And if they obey you then you have no cause against them. Indeed you have rights over your women, and your women have rights over you. As for your rights over your women, then they must not allow anyone whom you dislike to treat on your bedding (furniture), nor to admit anyone in your home that you dislike. And their rights over you are that you treat them well in clothing them and feeding them.”
(Jami' at-Tirmidhi1 2, Sunan ibn Majah and Musnad Ahamad)
Basically captives and slaves are "synonyms" in this case as in Islam legal slavery is restricted on prisoners of war. But as the further speech shows it is more likely to be taken metaphorical or as a kind of figurative speech.
The meaning of captive or even slave in this case is that the woman/girl after marriage moves to the custody and guardianship of her husband and that he has more rights over her than her father and therefore she has to obey her husband in first place.
And the major view of salafi scholars and in the Hanbali madhhab is that the wife has to serve her husband, while in the Hanafi, Maliki, Shafi'i and Dhahiri madhhab this service has some conditions and restrictions.
On sources of the statement
As for the sources ibn Tayymiyah attributed the statement to 'Umar ibn al-Khttab in his majmo' al-Fatawa مجموع فتاوى ابن تيمية (see here in Arabic between p 184 and 185) without sharing a source.
Sa'id ibn Mansur سعيد بنُ منصورٍ reported and attributed it in his Sunan to Asma' bint abi Bakr (hadith #591). And it should appear in Sunan al-BAyhaqi al-Kubra.
Note that in Shia' sources compile a very similar statement which is attributed to the prophet () see hadith #46/1139 here in Sheikh at-Tussi's al-Amali:
إنما النكاح رق، فإذا أنكح أحدكم وليدة فقد أرقها، فلينظر أحدكم لمن يرق كريمته.
See also fatwa from islamweb #27172 -in Arabic- on the explanation of some terms of hadith scholars. And islamweb #143181 on the authenticity and qualification of the discussed narration.
You may find this article explaining the athar also helpful!