The short answer is that the silent letters conceal hidden treasures. They weren't just randomly put there because our ancestors felt like it, but in every case they reflect actual pronunciation information from a former time. They still exist because people clinged to the old spellings while embracing new pronunciations.
One really cool word people like talking about is knight. You can see two distinct groups of letters that are silent in English. The first is the K, and the second is the GH. Both of these were actually pronounced, and you can see modern examples of this in the Dutch word knecht, in which the K and the N are distinctly pronounced, and in which the CH is pronounced in all its uvular glory. In the word Knight, we can't be sure of exactly how it was pronounced without recordings, but we can be very confident using reconstruction techniques, which include looking at other languages like Dutch and German. In Old English, people still pronounced the hard ch sound, which appeared as an H or even a GH, as in the word knight. People eventually figured it was easier not to say the K and the GH, but we still have them left in the spelling as remnants.
Another thing that people often talk about is the silent E at the end of words like bite and mate. This too reflects the fact that the E was actually pronounced earlier in the timeline. The transition to dropping the pronunciation of the E went along with the advent of the Great Vowel Shift, which transformed the way vowels work in English. You can see the difference if you look at the German language, where the E at the end of the word is almost always pronounced. For example, the translation of "I bite" into German is "Ich beiße." In German, "beiße" is two syllables, because the E at the end is pronounced like "uh". In English, the purpose of the E at the end of a word like bite has taken on a new role. It still signals a phonologically important piece of information; that the I is a long vowel instead of a short vowel (compare with the pronunciation of "bit").