Why does Equiano blame the illness aboard the ship on the "improvident avarice" of the traders?
The word "improvident" means "short-sighted" and "avarice" means greed.
So what Equiano is saying is that the slave traders were so greedy that they didn't think about the long term -- just about the short term. In this case, what that means is that they packed too many slaves into the ships because they wanted to make as much money as possible.
Because they did this, the living conditions for the slaves were terrible and many of them became sick. This was improvident because it didn't really take into account how much money the slave traders would lose by having the slaves be in bad condition when the time came to sell them (or how much they'd lose by having slaves die along the way).
The use of the terms "improvident" and "greed" implies that the slave traders weren't thinking too clearly. It is an interestingly understated way of perhaps suggesting that the slave traders were, in a sense, stupid. That's not something Equiano could get away with saying explicitly, especially not in the late 18th century.
Equiano is making the point that the slave traders were harming their own ability to make money off the slaves by subjecting them to such unhealthy conditions. He expresses the consequences of the slave traders' practice this way:
Many a time we were near suffocation from the want of fresh air, which we were often without for whole days together. This and the stench of the necessary tubs [for human waste removal] carried off many.
By forcing so many slaves into so small a space under decks they lost a number of them to illness and suicide.
We have heard many times about the horrific living conditions of the slave ships and the ocean crossing that the Africans were subjected to, but Equiano's subtle jabs at the white man's thinking is something a bit different.
Equiano goes on to shine a light on the white man's moral sense when he questions the need to break up African families:
Surely this is a new refinement in cruelty which, while it has no advantage to atone for it, thus aggravates distress and adds fresh horrors even to the wretchedness of slavery.
So, it isn't just the slave traders who make suspect decisions, it is also the slave owners. According to Equiano, they are destroying families simply out of the desire to be cruel, although it actually accomplishes nothing.
Taken from his village in what is now Nigeria, Olaudah Equiano was eleven when he was first captured. After six or seven months of captivity, he arrived at the sea and is horrified at the sight of
...a multitude of black [sic] people of every description, chained together, every one of their countenances expressing dejection and sorrow....
When he is moved into the hold of the ship, Equiano is overcome with the "loathsome stench,"and when he cannot eat, he is flogged. However, because the stench as the "whole ships's cargo were confined together" the area becomes what Equiano calls "pestilential." The confining environment, the nearly suffocating atmosphere of being packed together, added to the sweating and eliminating of the slaves, effected the death of many of them. This deplorable situation was "aggravated by the galling of the chains" and the screams of women and shrieks of the dying. Children even fell into tubs of defecation.
The "improvident avarice" of the slave traders, their desire to pack in as many as they can into the hold, along with their failure to maintain sanitary conditions lead to the death of many of the slaves. Fortunately for Equiano, because he is so young, he is not chained and can move onto the deck where he can breathe and get away from the gaseous smells.
He believed that by squeezing all the slaves onto the ships, they would become sick more easily, and therefore, the amount saved on space for the slaves would be smaller than the amount lost for the sick slaves.