Picture an input signal to an LTI system as being composed of an impulse train, where the individual strengths of the impulses are proportional to the instantaneous values of the signal.
The system responds to each impulse as and when it arrives, so the total system response at any instant of time will be the response to the present impulse plus all the remnants of the responses to the impulses that happened in the past.
The convolution sum is an intuitive graphical interpretation of this physical process.
Perhaps it's more intuitively appealing to time reverse ('fold') the input signal rather than the impulse response. This will produce exactly the same mathematical result, but the visualisation is then of the signal sliding through the system as time proceeds.