Using just multiplication could work if the filter is a simple gain that adjusts the amplitude of every sample by the same portion.
However, filters usually work on frequencies. For example a filter might be designed to remove (nearly) all frequencies above 8000 Hz. To do that work, it has to somehow figure out how quickly values are changing, on several different scales. However, multiplication by a constant or even by a vector of constants the same length as the signal, can never figure out how quickly values are changing and so cannot adapt the values being multiplied in order to deal with the frequencies.
Continuous convolution in theory gathers information across the entire signal, and applies the information learned at a particular time point. Discrete convolution does the same thing -- except that it happens to turn out that in some cases you can do a good-enough approximation by looking only at enough "nearby" samples instead of having to look at the entire signal.
