Why does SNR have an impact on the bit rate?
Yes, try to understand what is the concept of SNR  and it depends on what. Then, it will be easier to understand why SNR and BER increase and decrease in all cases.  

In your case, OFDM, the relationship between the SNR and BER is given: 

$\textrm{BER} = \frac{1}{2} \left(1 - \sqrt{ \frac{L/N \times \textrm{SNR}}{2 + L/N \times \textrm{SNR}}} \right)$

Where $N$ is the number of subcarrier and $L$ is the channel diversity. $\textrm{SNR} = p/\sigma^2$ .. where $p$ is the transmitted signal power, and $\sigma$ is the variance.

so now, you check the relationship between the type of used modulation with the above equation, (QPSK and N) , it's easier to explore the behavior of BER. remember also that probability of error when using QPSK is different about when using for example QAM and so on 

good luck