According to Digital Economist, indifference curves do not intersect due to transitivity and non-satiation. In order for two curves to intersect, there must a common reference point. That is impossible with indifference curves.
Transitivity means that consumers make rational decisions when they determine which good and how much of a good to buy. Non-satiation describes the assumption that consumers want as much of a good as possible. At a certain point that varies with each individual good, the preference for a good lessens. This consumption of a good is diminishing marginal utility. Consumers also typically purchase goods in groups. Instead of just buying 30 pounds of brown rice every day, consumers purchase multiple items. When consumers are indifferent toward the purchase of a group of items, the items give the consumer the same preference level. A hypothetical example is that consumers are indifferent toward the purchase of vegetables over fruit. If the price of one good changes, the budget line changes as well. This creates a change along the indifference curves, but the two do not intersect because the preference changes. The consumer decides to purchase less of an item, and the curve moves to the left. This is a rational approach to get as much of an item as possible.
Indifference curves are downward sloping. In addition to prices, taxes affect indifference preferences as well.