One way to think about real numbers is to see them as lengthes. In that setting, the product $a \times b$ represents the area of a rectangle whose sides are $a$ and $b$ (I'm not defining that precisely, but just relying on your intuition here). This explain easily why $a \times b = b \times a$.
Now in the special case where $n$ is an integer (and only in that case), we can check that $n \times x$ also has the meaning
$n \times x = \underbrace{x + \ldots + x}_{n \textrm{ times}}$
Indeed, it's easy to see why $(a + b) \times c = a \times c + b \times c$ (split a rectangle in two along the $a+b$ side) and $1 \times a = a$. So using this properties, we get
$2 \times x = (1+1) \times x = x + x$ $3 \times x = (2+1) \times x = x + x + x$ $4 \times x = (3+1) \times x = x + x + x + x$ $5 \times x = (4+1) \times x = x + x + x + x + x$
And so on.