I'm not sure how familiar you are with this but have you heard the term: like dissolves like? Well, anyway, this essentially means if you have a substance which is polar (such as the lithium bromide compound) then it will only dissolve in a polar solvent, like water.
However, with lithium bromide the lithium cation is rather small which means it can react with small ions, such as oxygen in an organic compound. Similarly, the bromide is a large anion which means that the two contrasting size ions don't interact to a great extent meaning the lattice can be broken easier. As a result, when the lithium cation reacts with the oxygen, for example, then the bromide has no choice but to follow due to their attraction. This means that lithium bromide is soluble in some organic compounds to some degree, anyway.
Hope I helped.