First of all, various implicit and explicit hints.
"In a fundamental manner": this is gesturing towards the so-called Fundamental Theorem of Arithmetic, which is the thing that says that integers have basically-unique prime factorizations.
"You'll need to take it a step further ... Everything must be squared!": when we have obtained some numbers in a reasonably straightforward way we will be squaring them.
"Red is the first, yellow the second. Blue is just another primary color." In keeping with the "fundamental" thing, this is clearly pointing at prime numbers.
OK, so what does this mean?
The most obvious thing to do, given the above, is to treat e.g. a square containing four red squares and one yellow square as denoting the number $2^43^1=16\cdot3=48$. This would give us [48, 16, 24 | 216, 72, 108 | 18, 6, 9]. I worried that the reference to blue as a third primary colour might mean that there are some blue squares in the boxes and we just can't see them against the blue background, or something, but fortunately this turns out not to be so.
square the numbers, getting [2304 256 576 | 46656 5184 11664 | 324 36 81]. But what now? You might e.g. hope for this to be a magic square, but obviously it isn't since the rows are of very different sizes.
something magic-square-like is true: if you take any row, column, or full diagonal then the sum of the numbers in those cells is a square. As is the sum of all the numbers in the 3x3 grid. (There are some other subsets that add up to squares to, but I think these are the specifically-intended ones.)