I frankly have no clue where you got this hadith. Perhaps give the original source from where you have got this translation from. In Bukhari, we only find this:
"The wife of the Prophet) The Prophet (ﷺ) used to practice Itikaf in the last ten days of Ramadan till he died and then his wives used to practice Itikaf after him." (Bukhari 2026, Kitab ul Itikaf, Book 33 Hadith 2)
Additionally, I tried my best to go through sunnah.com and as of yet haven't found this hadith as you have put it. Hence, to the best of my knowledge, it is quite possible that this is a fabrication.
Many atimes ahadith otherwise not found in the Sihah are quoted in tafasir. I've gone through a decent bit of tafasir under verse 2:187 (the longer verse about ramadan, itikaf, and marital relations in Ramadan, my guess as to where it might be present) and still can't find this hadith.
However, in my most humble opinion, the concept of this Hadith (even if it were true) doesn't contradict Islamic beliefs and seems to follow Islam's teachings on religion not being so strict when it comes to a legitimate necessity.
The Prophet (ﷺ) said, "Religion is very easy and whoever overburdens himself in his religion will not be able to continue in that way. So you should not be extremists, but try to be near to perfection and receive the good tidings that you will be rewarded; and gain strength by worshipping in the mornings, the afternoons, and during the last hours of the nights." (See Fath-ul-Bari, Page 102, Vol 1). Sahih al-Bukhari 39 Book 2, Hadith 32
and the Quran:
"O ye who believe! when you stand up for Prayer, wash your faces, and your hands up to the elbows, and pass your wet hands over your heads, and wash your feet to the ankles. And if you be unclean (junub), purify yourselves by bathing. And if you are ill or you are on a journey while unclean, or one of you comes from the privy or you have touched women, and you find not water, betake yourselves to pure dust and wipe therewith your faces and your hands. Allah desires not that He should place you in a difficulty, but He desires to purify you and to complete His favour upon you, so that you may be grateful." (5:6)
and
"No blame lies on the weak, nor on the sick, nor on those who find naught to spend, if they are sincere to Allah and His Messenger. There is no cause of reproach against those who do good deeds; and Allah is Most Forgiving, Merciful." (9:91)
and
"He has made unlawful to you only that which dies of itself, and blood and the flesh of swine, and that on which the name of any other than Allah has been invoked. But he who is driven by necessity, being neither disobedient nor exceeding the limit, it shall be no sin for him. Surely, Allah is Most Forgiving, Merciful. (2:173)
Islams teachings are that religion should not be overly-burdensome on anyone. If anyone has a legitimate reason (sick, injured, poor, etc.) then no blame lies on them for failing to abide by a specific Quranic commandment. Islam permits Muslims to even eat pork if they have to out of necessity.
So personally I think this alleged Hadith's meaning makes sense. If Aishah (ra) had a real necessity (which we would assume due to her blessed character) then it's no problem if she does something that would otherwise violate rules of Itekaf.
However, as explained earlier, I think it's most likely a fabrication since I cannot (to the best of my abilities) find any reference to it.
