An average is a summary of different values: if you have several signal processing exams, you know that both your answers and the grading person may incur some variability. On top of that, it is complicated to compare student, or the same student at different years, based on a series of grades.
So an average of grades both:
- compacts different grades in one single number,
- balances exam variability.
The latter is of first importance in signal processing: a signal sample may be disturbed by noise, sensor fluctuations, quantization, etc. While the signal is supposed to have some regularity, the disturbances are often chaotic, incoherent, from one sample to the other. If you average the distances alone, their amplitudes tend to diminish (not in all cases, but let us say that for now). So if close sample values are averaged, the ones from the signal don't change a lot (like a regular working student), while the disturbances reduce. Thus, the signal looks smoother, more pertinent. This is the basis for the linear filters: further, you can weight the samples differently.
The summarizing property is also important. You can average three samples, and keep the average only, and switch to the other sample triplet. This reduces the quantity of information, and is the basis for subsampling, data compression, etc. JPEG uses a lot of filtering and subsampling.
Both combined are the root of filter banks, putting several filters in parallel with subsampling. Then, you have nonlinear filters, but this is a whole new story.