In one interpretation of the convolution of two functions, one of them is interpreted as the input signal to a system, and the other the impulse response of that system. Therefore, the convolution gives the system's response to the input. See Section 17.4.1 in this book: https://books.google.com/books?id=q5KRCwAAQBAJ&pg=PA109&dq=zorich+mathematical+analysis+vol+I&hl=en&sa=X&ved=0ahUKEwilx9yf0OfQAhXrrVQKHdbFBdkQ6AEIJzAC#v=onepage&q=convolution&f=false
As a special case, the convolution of a function $f(t)$ with the Dirac delta function gives $f(0)$ (the response of the system at $t=0$).
Because of the above, the input signal is thought of as "smearing" the impulse response on the time axis. Mathematically, it means that convolution of a given function $f$ with a smooth function (a "smooth smearer") results in something smoother than $f$. This type of smoothing is extremely useful when you want to use differentials.
It also turns out that, on mapping from the time domain to the frequency domain (using a Fourier or Laplace transform), convolution of time-dependent functions maps to the product of frequency-dependent functions.