Relationship between Bandwidth and Bit Rate
Assume you have a pulse of width $T$ which is your bit.

$$w(t)=1\quad for\quad-\frac T2\le t\le\frac T2\quad and\; 0\;elsewhere$$
 Your bit in frequency domain is:

$$W(f)=Sinc(T.f)$$

Look up Sinc function and you can see $Sinc(T.f)$ main lobe is between $\pm \frac 1T$. As your bit is widened in time, it occupies less bandwidth in frequency domain($T$ increases, $\frac 1T$ decreases) and vice versa. This is a very important result to understand and remember for the rest of your life. It's the reasoning behind relationship of frequency $BW$ and pulse duration $T$. Carrier frequency has nothing to do with it, it just shifts your entire signal spectrum up in frequency to make it suitable for transmission in the physical medium(channel) of your choice.

You should review Fourier transform and its properties.  Then look up Convolution theorem, which tells if you multiply signals in time, they get convolved in frequency domain. Then learn about Sinc function and its relationship to Rect function.