I see this term thrown about with no regard to what it means and I thought I would do an OP to clarify what it really means. In order to to this I must first explain how things are done in physics. The first point to address is how do physicists find anything out in the first place and the answer to that is experimentation, this can be measuring something, seeing how something interacts with something else, a whole myriad of possible techniques.

Once the data has been gathered than the next task it to see what information can be obtained from the data, as well as if it possible possible to codify the data in the form of an equation, such equations are usually known as laws. The more experiments performed more laws are obtained and then the physicist can start incorporating the laws to make more general laws that explain more of the experimental onservation. These new more general laws will in general have the ability to predict the outcome of experiments that have yet to be performed and can be used to test the robustness of the laws themselves.

These general laws can sometimes be made into more general laws which can describe a whole range of phenomena, these more general laws are known as a theory in physics.

What most people refer to as the "laws of physics" are these sets of equations that have been discovered by physicists which describe (in the majority of cases to a very high degree of accuracy) what appears in nature.

It is not that nature appears to be inherently mathematical in nature but that it can be approximated by mathematical laws to an incredibly high degree.