Disclaimer: It's been awhile, since I used step functions, so read my comments carefully because I might not know what I'm talking about.
How are they multiplied?
u[n] * u[n-3]
This multiplication?
I'm not sure what the engineers might be doing (i.e., signal systems), but I'm thinking that the expressions u[n] and u[n - 3] are each symbolic numbers (function notation) that represent output values of step-function u, when the inputs are n and n-3, respectively.
In other words, the product that defines function y is already simplified. I don't think that we can multiply that expression.
If you can tell us what the actual definition is for function u, then perhaps we could multiply u[n]*u[n-3] using their definitions, and simplify this product in terms of n.
To determine the piecewise definition for function y that you posted as the answer, I'm thinking that you must have more information about function u.
Are you familiar with the Heavyside function? That's the basic unit step function.
H[n] = {0, n < 0; 1, n >= 0}
So, I'm thinking that your u[n] is similarly defined, except that the jump discontinuity is shifted three units to the right (i.e., at n = 3) and further that u[n] is defined in terms of n, when n is three or more. But, this is kinda a guess.
u[n] = {
0, n < 3;
[not sure what goes here], n}
This is why y[n] is defined as zero, when n<3, yes? In other words, u[n] is zero whenever the input n is to the left of the jump discontinuity at n = 3. And, whenever n is to the left of 3, the input n-3 is clearly also to the left of the jump discontinuity; hence, u[n-3] is also zero when n<3. That makes y[n] = 0*0, when n is to the left of the jump discontinuity.
The part that I cannot figure out is why y[n] starts growing to the right of the jump discontinuity (it's growing because y[n] is defined in terms of n when n>=3; that is, as n gets bigger, so does y[n]). I just can't remember a unit step function that has a slope other than zero to the right of the jump discontinuity.
Hopefully, though, my comments show that, at least for n<3, we can
analyze the expression u[n]*u[n-3] without doing any multiplication. Maybe, you can now reason why y[n] is n-2, for n>=3.
Do you have more information about the step function u[n]?
Again, if I'm way off base in this post, feel free to ignore it.
Cheers.