Hello Mr. wolfheart
I am not sure if it is too late, but this was and to a certain extent still is a question that bugs me, but after a couple of years I have come to get a reasonably good feeling for what the Fourier transform is doing.
First off, I assume that you know that the complex exponential is just a fancy way of writting cos(x)+i*sin(x). This is from euler's formula. This is easy to look into on google. Anyways, when people say functions are orthogonal to other functions, they are esentially saying something very similiar to when you say one vector is orthogonal to another vector. I am going to assume that you have some kind of engineering background based on the fact that you even care about this topic so I assume you know what the dot product is for vectors. Basically when we dot product two vectors together you are multiplying the x components together, the y components together, and the z components together, and then adding together these products.
dot product = x1*x2 + y1*y2 + z1*z2
You are essentially doing the same thing with the fourier transform. You can think of the complex exponential, and the signal under study as the two vectors. You can think of every instant in time as the x,y, z, components (except it is no longer discrete in the same way the x,y,z used to be) The integration is just a fancy sumation of all the products of the two functions (vectors). This is how I generally think about it. You are essentially dot producting tw vectors. Remeber when you do a dot prooduct with vectors you are esentially trying to see how well the two vectors line up. When they are perpendicular to each other they don't line up at all and the result is zero. when they line up perfectly, the result is simply the product of the magnitudes of the vectors. Everything else is somewhere in between.
There is another way to look at this too. If you are comfortable with the concept of correlation the fourier transform is essentially just trying to find out how well a signal is correlated with signs and cosines.
For more informaiton, I think one of the easiest websites to understand is wikipedia. lookup correlation cross correlation, fourier transform, signal processing, etc.
Personally I think oppenhiem's book is very hard for begineers to understand. Try looking in a signal and systems book such as "signals and systems" by Chi-Tsong Chen.
I hope this helps and I would be curious to know if helps or not. I have spent years trying to get a feel for this thing and alot of the theory generally is hard to get a feel for. I think most people are more comfortable thinking in terms of vectors.