As Glen_b noted in comments, think of a Poisson variable as counting the number of events for a Poisson process of intensity $\alpha$ in a fixed window of size $w$: the Poisson parameter for this variable is $\alpha w$.
Suppose now there are two independent Poisson processes separately running in the same window, one of intensity $\alpha_1$ with $\alpha_1 w = \lambda_1$ and the other with intensity $\alpha_2$ with $\alpha_2 w = \lambda_2.$
This figure shows realizations of two independent Poisson processes on the line within the window displayed as a gray line segment: one is shown as the orange points and the other as blue points. Their counts are realizations of two Poisson variables. By ignoring the colors, as shown at the bottom, we obtain a realization of a Poisson process whose intensity is the sum of the original intensities. The count is the sum of the counts.
The sum of these counts clearly is a Poisson process of intensity $\alpha_1 + \alpha_2$ because it trivially satisfies all the requirements of Poisson process (the count is proportional to the size of the window, events in non-overlapping windows are independent, and the chance of a count exceeding $1$ grows vanishingly small as the window size shrinks to zero). Therefore the sum of counts is a Poisson random variable with parameter $(\alpha_1+\alpha_2)w = \lambda_1+\lambda_2,$ QED.