Why convolution in time domain is multiplication in frequency domain?
Not remotely a mathematical proof, but I gather from the post that you understood the math and wanted a more intuitive understanding of the relationship...

When you convolve, you create a time series, but the time axis isn't real time, it's time difference between the two time-based functions.  You then integrate the product across the time axis for each time difference to get your result in the time domain.

A time difference is a period, and the inverse of period is frequency.  Each point in X(f) represents a frequency integrated across the time axis.  In the frequency domain, therefore, X(f) and Y(f) have already been integrated with a reference function across the time axis.

Now that that integration is complete, the multiplication is the remaining operation.  Since you've already integrated across the time axis, multiplication gives you your result...but it's still in the frequency domain.  An inverse Fourier transform is still needed to get the time result.

Interestingly, it's symmetric...multiplication in time is convolution in frequency.  But we invented a different word for it...heterodyne.