But what will be meant by "per unit change" in this case? As the proportions lie within [0,1], I am getting a little confused with what a "per unit change" will mean in this scale. Does it mean 0.01 or 1%? (I am sorry for my noob thoughts!)

In my data the range of the proportions is 0 to 1, not multiplied by 100. Do I need to multiply them by 100? So that I can say "per unit change" means 1% change? I have seen that the coefficients do differ in scale if I multiply the proportions by 100. For example, a coefficient of -1.3 for proportions becomes -0.013 for percentages (when the proportions are multiplied by 100).