You're right, this doesn't make as much intuitive sense as the substitution effect. But here goes:
For most things, when their price goes up, you don't feel particularly poorer. That's true because most things don't really take up much (a very high percentage) of your income. For those types of things, the income effect really doesn't exist.
However, there are some things that are really expensive relative to your income (for most people). Housing is one of them. If your rent goes up 10%, you are likely to feel poorer because it's actually a lot of money compared to if the price of a Coke goes up 10%.
So that much makes sense to me. My mortgage is about $800 a month and if it went up to $880 a month I'd notice and probably feel poorer.
As far as who said it's true... I don't know that one. Just googling it, I've found a couple papers already that argue that it doesn't actually exist... at least not separate from the substitution effect.
Perhaps, income effect does not make much sense, because the way it operates is not very obvious to us. Unfortunately what makes sense to us may lead us to wrong conclusions, because things are not always as simple as they appear to us.
First thin to clear this confusion about income effect is to understand that the income effect relates to "real income" rather than the of money or dollars. Thus when the prices of goods increase, I will be able to by less of at least some of the goods with the fixed amount of money I have. To make it simple to understand just take example of one prices of just one product. Let us say that Out of monthly income of $3000 a family spends $100 on bread. This leaves $2900 to be spent on all other things. If the price of bread is increased by 30%, the company will need $130 to buy the same quantity of bread. Now let us assume that bread being an essential commodity for the family, they continue to buy and consume the same quantity of bread as earlier. This leaves the family only $2870 to buy all the other goods. Thus their real income has reduced by $30 or by 1% of total income. The family may not feel the pinch of just 1% reduction in their purchasing power, but it has definitely reduced because of income effect of rise in prices.