exp(-t/tau) is an exponential function of t, whose value is 1 at t = 0, whose value is less than 1 for t > 0, and whose value is greater than 1 for t < 0.
Here's a plot of exp(-t/tau) without the heaviside(t):
syms t real
tau = 0.1;
f(t) = exp(-t/tau);
fplot(f,[-1 1])
Notice the very large values f(t) has when t < 0.
Look at those same values on a log scale:
fplot(f,[-1 1])
set(gca(),'YScale','log')
Notice the function goes through 1 at t = 0.
In order to have the function start at t = 0 and discard the values for t < 0, you can use the heaviside() function, which has value 1 for t > 0 and value 0 for t < 0 (its value is 1/2 at t = 0):
fplot(heaviside(t),[-1 1],'LineWidth',2)
Multiplying exp(-t/tau) by heaviside(t) has the effect of zeroing-out the function for t < 0:
f(t) = exp(-t/tau)*heaviside(t);
fplot(f,[-1 1])
