I am sorry for how rudimentary this question will sound.
I approach the frequency domain thinking in discrete terms. The plane is frequency on the x axis and amplitude on they y (ignoring phases). Each spike represents a complex sinusoidal that, added with the others, will approximate the original curve in the time space.
For example:
Is the sum of two complex sinusoidals that will generate a real cos(200Pi). (Picture taken from here : https://ccrma.stanford.edu/~jos/mdft/Sinusoid_Magnitude_Spectra.html )
What I don't understand is the move to Fourier Transforms and how suddenly I have continuous curves in the frequency plane (for example the Fourier Transform of a Gaussian function) becomes something like this:
Now I am really at a loss as to how to understand the meaning of a continuous curve in that space. Is there a way to understand it as a sum of distinct complex sinusoidals?
The first thing to understand is that any periodic function can be represented as the sum of discrete "spikes" in the frequency domain, that is, by its "Fourier series". The longer the period of the function, that is, the longer our sampling interval, the closer together those spikes need to be.
In that vein, one common explanation is that the Fourier transform is the limit whereby we take Fourier series of functions over progressively larger sampling intervals. We go from a sum of discrete sinusoids divided by $N$ to an integral of sinusoids multiplied by the "infinitesimal" $dx$.
In a sense, we are still just "summing" sinusoids, but those sinusoids are infinitely close together in the frequency domain.
See page 8 here for a more explicit treatment.