First of all, the letters did not become silent as letters are not linguistic units and the orthography of a language is quite arbitrary. In fact in French, there was a time when linguistists (or precursors thereof) were tracing Latin origins of the words and in order to make French closer to Latin, they renewed letters for phonemes that were lost long ago, like Latin "tempus" (time), which gave French "tens" (today just [tã] and reflected in the English word "tense" as in "grammatic tense") but was later changed by the grammarians to "temps" (this was sometimes done incorrectly, e.g. French "pois" - a "weigh" came from Latin "pensum", but grammarians traced it to Latin "podus" and add the letter D into the word, resulting into today's "poids").
So if we modify your question why so many original latin phonemes disappeared, the answer is fairly simple:
Ancient French, probably under the influence of Germanic invaders, developed a strong dynamic accent (compared to Latin melodic). This led to the same effect as in today's English, where accented vowels are reinforced (often diphthongised) while non-accented vowels are reduced (typically to schwa). In French, this led to loss of all post-accent syllables, which combined with the general Romance tendency to drop final consonants or consonant clusters led to what the French looks like today.
However since there were contexts where some of the consonants were still pronounced (e.g. before a vowel of a following word - today's liaison), they were kept in writing mostly ("il finit" > "finit-il?" also compare with "il parle" > "parle-t-il?", which is the same phenomenon, just the original Latin T for 3SG was dropped for the verb class entirely).