which are you confused about? the integration or the y = ... or both.
The TA did integration by parts to find the integral, and to go from
e^(3t)y = (1/3)te^(3t) - (1/9)e^(3t) + e^t + C
y = (1/3)t - (1/9) + e^(-2t) + Ce^(-3t)
he divided both sides by e^(3t)
which parts exactly do you want explained? do you know the method for integration by parts?