Direct sequence spread spectrum modulation When the receiver gets the signal $m_t$ sent by the transmitter, the signal will have an additive interference $i_t$. Thereby signal received will be : $$r_t = m_t + i_t$$

I know that to get back the data from the signal, we have to multiply by the pseudo-noise sequence (PN). 

That will take out the pseudo-noise from the signal $m_t$ allowing us to get the data.

What do we do with the noise $i_t$?

It has been multiplied by the PN but I supposed we cant take out the data from the $r_t$ until we eliminate the noise. I don't really see what kind of signal do we get multiplying the additive interference by the PN. I've read that we use an low pass filter but i dont really undestand why.
Multiplying $r_t$ by the pseudo random spreading code does different things to $m_t$ and $i_t$: Because $m_t$ is correlated to the spreading code, it is despread while keeping its power, so the power spectral density will be higher in a small bandwidth (due to remaining modulation). On the other hand, $i_t$ is not correlated and is translated to wideband noise again.

This gives you the opportunity to filter with a bandpass and just use the band, where the signal to noise ratio is above unity.

The noise is not eliminated, but its overall power is reduced, because your bandpass rejects some of the noise (that is still wideband) and passes all of the signal (that is now narrowband).