please explain this fourier equaton
This equation contains ALL fundamental frequency terms. It's written to find how a's and b's are connected to c's.

In the red box the left side is a general formula how to make an arbitary real valued sinusoidal signal with arbitary phase angle by summing sine and cosine which both have ZERO phase angle.

In the right side there is a sum of complex numbers to get rid of the imaginary part. This is the start of the derivation the relation, so it's not said that the right side must have total imaginary part=0, but it will surely be said later.

Hopefully you know Euler's identity. Without it this stays as a piece of mumbojumbo.