What does it mean to expand a function in its basis?

From a physical point of view, you are always interested in observables, which you describe quantum mechanically by means of some hermitian operator. In fact, a property of Hermitian operators is having real eigenvalues, which correspond to the allowed measurement outcomes for a measurement of the observable of interest. Each of these measurement outcomes is, as an eigenvalue, associated to an eigenstate and the set of all of these forms an orthonormal basis. Physically, these eigenstates describe the post-measurement states for a measurement of the corresponding eigenvalue. In your case, the set of eigenstates ${|omega_{i}rangle}_{i}$ may be associated to some observable described by the hermitian operator $Omega$ with eigenvalues ${omega_{i}}_{i}$. Expanding the state $|psirangle$ describing your system as a superposition of these eigenstates is useful to determine the probability for each measurement outcome $omega_{i}$, indeed given by $|langle omega_{i} | psi rangle|^{2}$, the absolute square of the coefficients in the expansion you wrote.

I was reviewing my quantum mechanics notes, and I was confused about what this expression meant:

First of all, given a vector $|psirangle in mathcal{V}^n(mathcal{C})$ you can always expand it in term of indepedent basis set. Let us suppose an orthonormal complete basis set given by ${|phi_irangle}$. $$|psirangle=sum_ic_i|phi_irangle$$
To find coefficient we can take dot product $$langlephi_i|psirangle=c_i$$ putting everything together we get $$|psirangle=sum_ilanglephi_i|psirangle|phi_irangle=sum_i |phi_iranglelanglephi_i|psirangle =left(sum_i |phi_iranglelanglephi_i| right)|psirangle$$

That explain the math. One point to note that the object on the right in identity ( a complete set of basis) so whenever we need to do expansion or change of basis etc. we insert a complete set of basis.

Now let's see the physics. To see the implication of the concept we just need to recall the postulate of quantum mechanics. Here I'm writing those which are important for discussion.

If the particle is in a state $|psirangle$, measurement of the variable (corresponding to) $Omega$ with probability $P(omega)propto |langleomega|psirangle|^2$.The state of the system will change from $|psirangle$ to $|omegarangle$ as a result of the measurement.

Just reading this two times give you reason of above math. One thing to note that basis set is not unique, So one always choose the eigenbasis (correspond to operator of interest) when doing measurement.

Hope this help you. Best wishes!