The saying is often attributed to St. Francis Assisi. It basically argues the idea that Christians spend too much time talking about Jesus and not living like Him, which has its merits to be sure, but one of the things Jesus did during His time on Earth is proclaim the kingdom of God--out loud, with words, using his mouth, talking to people. He did not pull His punches. He was not politically correct; in fact, He was deeply offensive, so much so that those who disagreed with Him killed Him.
And just to be clear, St. Francis Assisi probably didn't author the quote above, and if he did, he certainly didn't mean that Christians ought to solely use their "silent testimony" and never share the Gospel out loud lest they offend someone. How do we know?
We know because St. Francis Assisi was a fiery preacher that condemned sin and preached the Gospel. He said crazy things like “repent, performing worthy fruits of penance, because we shall soon die . . . Blessed are those who die in repentance for they shall be in the kingdom of heaven. Woe to those who do not die in repentance, for they shall be children of the devil whose works they do and they shall go into everlasting fire." And then he sent his followers out as missionaries to proclaim this message, not really a silent, “look at my actions” thing to do.
And most importantly, let’s not forget that the Gospel is good news. Unless you’re a mime, how do you share news without using words to communicate it? And without the good news of Jesus Christ, how can anyone be saved?
So, pursue social justice: feed the hungry, clothe the naked, and champion the powerless; live out your faith: flee from sin, love your neighbor as yourself, pray for the sick, and obey the commandments; but don’t do it with your mouth shut: preach the Gospel, proclaim the kingdom of God, and give glory to God for any good that comes out of your life because He is worthy!
How then shall they call on him in whom they have not believed? And how shall they believe in him of whom they have not heard? And how shall they hear without a preacher? (Romans 10:14)