I solve this differential equation using Mathematica. But I don't understand the solution.
Why the lower limit of this integral is 1?
I run: $$\text{DSolve}\left[y'(x)+y(x)=Q(x),y(x),x\right]$$ the solution: $$\left\{\left\{y(x)\to e^{-x} \int_1^x e^{K[1]} Q(K[1]) \, dK[1]+c_1 e^{-x}\right\}\right\}$$
If you're going to express the solution using a definite integral, that integral needs endpoints.
Since no initial condition was specified, there is no reason to choose $1$ as the lower limit of integration. The choice is completely arbitrary. I have no idea why Mathematica makes this choice. But it doesn't matter. A different choice of lower limit would just change the arbitrary constant $c_1$. So you could just as well write the solution as $$ y(x) = e^{-x} \int_{2013}^x e^t Q(t)\ dt + c_2 e^{-x}$$