You are right that relevance logic is motivated by a desire to obtain a more natural account of conditionals and to exclude as true those where there is no relevant connection between the antecedent and consequent.
A fairly standard approach to the semantics of the conditional of relevance logic is to understand it in an information-theoretic way. It can be thought of as expressing an information channel between two information sites, or as an implication between situations under various restrictions, or as a conditional that is subject to pragmatic maxims, or as a relation that combines information to yield a result.
The sentence, "if the moon is made of cheese then 2+2=4" holds true for the material conditional and also for the strict conditional. In relevance logic, the antecedent is irrelevant to the consequent, so it does not hold. We might understand this by observing that the information expressed by "the moon is made of cheese", whether it holds or not at a particular world, contributes no information to the fact that 2+2=4 holds, so it is redundant.
In order to understand what 'information' and 'contributing information' mean in this context, you would need to go into the details of Routley-Meyer semantics and the varieties of relevance logic that arise based on the restrictions imposed by its accessibility relation.
The conditionals of classical and intuitionistic logics do indeed produce many odd results. So much so that they cannot be considered general accounts of the logic of conditionals. Natural language conditionals are often non-monotonic, context-dependent, non-truth valued, uncertain, etc. Also, it is important to remember that conditionals are not just used to conditionalise statements. We can conditionalise questions, commands, offers, threats, bets, promises, etc. A satisfactory account of conditionals should be able to include these also, though many make no attempt to do so.
There is a huge literature on conditionals that attempts to understand their logic, their semantics and their pragmatics. Some approaches are: the use of non-classical logics such as relevance logics, probability logic, connexive logic and default logics; the addition of stronger conditionals to classical logic such as strict and variably strict conditonals; theories of counterfactuals; non-truth valued accounts of conditionals; evidential theories of conditionals, linguistic treatment of conditionals as quantificational restrictors; dynamic understanding of conditionals in terms of belief revision.
If you are fairly new to conditionals, it is worth reading some of the articles in the Stanford Encyclopedia, particularly on Indicative Condiitonals, Counterfactuals Conditionals, and The Logic of Conditionals. A good introductory book is Jonathan Bennett's Conditionals: A Philosophical Guide (2003).
