1) Orthogonal matrices arise naturally when working with orthogonal bases or orthogonal transformation. Working with orthogonal bases is very handy because it allows you to use formula like Pythagoras or it allows you to work with Fourier series.
2) Orthogonal matrices have a great numerical stability. Multiplying with an orthogonal matrix causes almost no errors. Furthermore, there are a lot of decomposition theorems involving orthogonal matrices. For example the singular value decomposition.
3) Orthogonal matrices correspond to the linear isometries. So from a categorical point of view, they are the isomorphisms of normed vector spaces. We often identify between spaces if they are the same up to linear isometry.