An association rule is an an implication of expression of the form $X$ results in $Y$ and the strength of an association rule can be measured in terms both values:Support and Confidence. Both of them are great metrics but can cause some confusion. Support is important because a rule that has low support may occur by chance. Confidence measure the reliability of the inference made by rule. In the other words Lift give you a better insight about the importance of a rule.
Lift shows how much an antecedent increase the probability of the consequence. We can say if a lift is greater than 1 the antecedent positively affect the consequence. The more is better, we can say , for example, if lift is 1.25 the antecedent increase the probability of occurrence of the consequence by 25%
In your example, you can say the combination of the right hand and left hand items is the whole basket and support indicates the most popular baskets. But again be careful about the interpreting the results. You might have some baskets with high supports with low confidence. It means, although the left hand item happened many times with the right hand one, it is very likely that the left hand one occur with the other items, as well.
I don't go to mathematics behind them since they are very simple.