I think the steps would be like this:

- get understanding of ordinary vector linear algebra.

- understand what vector dot product does and why

- understand why an orthogonal set of basis vectors for the space you're working in is useful / what properties it has / how its used. like basic euclidean 3d space (1,0,0) (0,1,0) (0, 0, 1) basis vectors.

- get a refresher on basic calculus, in particular integrals

- understand this inner product, it's a generalization of dot product, except you can think of your vectors having infinite number of dimensions now.

- the properties of the dot product you know (like that two vectors are perpendicular if their dot product is 0) work for the inner product too. or perhaps its better to say that the general inner product is defined to have similar properties

- there are functions that are orthogonal to each other in the same way vectors can be orthogonal to each other, and you can use the inner product to tell which ones.

- spherical harmonics are constructed / by design orthogonal to each other. how to show this and where the intuition for finding them could come from is a whole topic...

- but once you have it, just like you can project vectors onto basis vectors (to essentially transform them into the coordinate system described by those basis vectors), you can project functions into the coordinate system represented by those orthogonal functions.

- then you have to figure out why you would even want to do this. in short is has a lot of useful properties/applications. in the graphics case you can compress some quite complex functions into just a few coefficients using this (not perfectly, there is some 'information loss', but still). integrating over two functions becomes cheaper when they are projected to SH basis. it lets you do some unintuitive stuff like combine light that goes into different directions into one common set of coefficients.

Introduction to spherical harmonics for graphics programmers