Answer
When you multiply two polynomials, you multiply each of the terms of the first polynomial by each of the terms of the second. Thus, the highest power term in the first polynomial (the one that determines the degree of the polynomial) is multiplied by the highest power term in the second polynomial once in the process of multiplying the two polynomials. Since the two highest degree terms are nonzero multiples of powers of the variable (say $x$ for example), and since multiplying powers of the same variable is the same as adding the powers, the degree of the two polynomials multiplied is the degree of the the first added to the degree of the second.
Work Step by Step
Assume you have two polynomials $y_{1}$ and $y_{2}$ of x:
Since these are polynomials, they are of the following form:
$y_{1} = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + ... +a_{n}x^{n} $ for some natural number n
and
$y_{2} = b_{0} + b_{1}x + b_{2}x^{2} + b_{3}x^{3} + ... +b_{m}x^{m} $ for some natural number m
(where $a_{0},a_{1},...,a_{n}$ and $b_{0},b_{1},...,b_{m}$ are all real numbers)
Thus, when they are multiplied:
$ y_{1} * y_{2}$ = $ (a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + ... +a_{n}x^{n})*(b_{0} + b_{1}x + b_{2}x^{2} + b_{3}x^{3} + ... +b_{m}x^{m})$ = $(a_{0}*b_{0} + ... + a_{n}x^{n} *b_{m}x^{m})$ = $(a_{0}*b_{0} + ... + a_{n}*b_{m}x^{m + n})$
$\implies$ the degree of $y_{1} * y_{2}$ is $m+n$
