The work of physicists is to construct mathematical models that describe the world around them. This sounds complicated, but a mathematical model is just a set of equations. For example Newton's laws, as learned by generations of schoolchildren, are a mathematical model to describe motion.
Maybe this is getting excessively philosophical but the key point of a mathematical model is that it's a description of reality. No-one is claiming the model is really what happens (whatever "real" means) - it's just a description. You also need to bear in mind that all mathematical models are approximations to reality, because they make simplifying assumptions, and they have a limited scope i.e. there will be limits beyond which the model no longer applies.
A good mathematical model will usually answer lots of "why" questions. For example you might ask why a neutron decays. Well you can use the mathematical model called the Standard Model and this will tell you why the neutron decays. In fact it will answer lots and lots of why questions. But suppose you ask why is the electron mass 0.51MeV and the Standard Model can't answer this because it's put in as an assumption when constructing the Standard Model. If you ask a question about the electron mass you've reached the limits of the mathematical model and you need an improved one.
For an improved mathematical model we might take String Theory, which (hopefully) extends the Standard model to answer lots more questions. When we understand String Theory fully we can hopefully tell you why the electron mass is 0.51MeV. But there will be other deeper why questions that even String Theory can't answer, and for those we'll need an even better model.
So your distinction between how and why questions is a bit artifical. Physicists are interested in why questions, but only if they can construct a mathematical model that can address them. Whether there are why questions that cannot be answered is an open question. Einstein famously hoped general relativity would prove the universe had to be the way it is because it had no alternative. In this he was disappointed. We'd all love to have a theory of everything that would answer every why question but at present we are far from this.