Trouble interpreting Bode diagram

Gauss M.D.

Trying to get my head around frequency response and Bode plots, but I'm having trouble interpreting what's going on. The bode plot of a function is expressing what would happen to my function if I multiply it with sin(ωt), right?

To get a better feel for it, I asked Matlab to give me the Bode plot of 1/(s^2+1), ie sin(t). Shouldn't I be getting a graph of the different amplitude ratios for sin(t)*sin(ωt)?

But according to Matlab, the amplitude ratio approaches infinity at ω = 1. I don't get it. Can anyone explain what I'm not getting?