Integrator and the multiplier in the Optimal Binary Detector
There are several things missing/extra in your diagram. 


What you are using is 
rectangular PAM pulses of duration $T$ to send data across the channel, and so 
you really don't need the multiplier. It is necessary only if $s_1(t)$ and $s_2(t)$
are different from rectangular pulses (though they are still of duration $T$, 
and in that case, the input $s_1(t)-s_2(t)$ to the multiplier does not tell 
the whole story. 
The integrator that you are using is really what is called an 
integrate-and-dump correlator. The input is 
integrated over intervals of time $[(n-1)T, nT)$. Just before the end of
the integration period, at time $nT^{--}$, the integrator output is sampled
and the sample value is what is used in the threshold device. Following
the sampling at $nT^{--}$, the integrator is dumped (meaning that 
the output is reset to $0$) at time $nT^{-}$, and the integrator is
restarted at $t = nT$ for the next data bit from $nT$ to $(n+1)T$.
Note that there is what computer engineers call a critical race involved:
the circuit designer has to make absolutely sure that the sample signal 
arrives at the integrator before the dump signal. Getting this wrong
can be a fireable offense.
With the integrate-and-dump correlator working as described above,
the input to the multiplier during $[(n-1)T, nT)$ needs to be
$s_1(t-(n-1)T)-s_2(t-(n-1)T)$, that is, the signal $s_1(t)-s_2(t)$
(which is nonzero only for $t \in [0,T)$) delayed by $(n-1)T$ seconds.
What the integrate-and-dump correlator is doing is computing the
output of the matched filter for signals $s_1(t)$ and $s_2(t)$
in additive white Gaussian noise at times $nT$.  Note that the
correlator output is not equal to the matched filter output
at times other than $nT$.
The reason you cannot simply sample the received signal
and make a decision from the sample is that the SNR is abysmal.
Technically, AWGN has infinite variance 
(cf. this question),
but even otherwise this is a big problem.  The matched
filter maximizes the SNR at the sampling instants,
and the correlator is just calculating what the matched
filter will give you at the sampling instants.


For details of all this, see, for example, pp. 85-93
of these ancient lecture notes of mine.