Say we are using the example of a grocery store that is testing the validity of an association rule that has an antecedent and a consequent (for example: "If a customer buys bread, they will also buy butter").
If you look at all transactions, and examine one at random, the probability that that transaction contains the consequent is "Expected Confidence". If you look at all transactions that contain the antecedent, and select a random transaction from these, the probability that that transaction will contain the consequent is "Confidence". "Lift" is essentially the difference between these two. With lift, we can examine the relationship between two items that have high confidence (if confidence is low then lift is essentially irrelevant).
If they have high confidence and low lift, then we still know the items are frequently bought together but we do not know if the consequent is happening because of the antecedent or if it is just a coincidence (perhaps they are both purchased together often because they're both very popular products but don't have any kind of relationship to one another).
However, if the confidence and lift are both high, then we can reasonably assume that the consequent is happening due to the antecedent. The higher the lift gets, the lower the probability is that the relationship between the two items is just a coincidence. In mathematical terms:
Lift = Confidence / Expected Confidence
In our example, if the confidence of our rule was high and the lift was low, that would mean that a lot of customers are buying bread and butter, but we do not know if it's due to some special relationship between bread and butter or if bread and butter are just popular items individually and the fact that they often show up in grocery carts together is just a coincidence. If the confidence in our rule is high and the lift is high, this indicates a pretty strong correlation between the antecedent and the consequent, meaning that we can reasonably assume that customers are buying butter because of the fact that they are buying bread. The higher the lift is, the more confident we can be in this association.