Each of the $10$ balls can choose which one of the boxes to go to. Since there are $4$ boxes, the ball has $4$ choices. This is true for each of the $10$ balls, so the product rule tells us there are $$\underbrace{4 \times \cdots \times 4}_{10 \text{ times}} = 4^{10}$$ ways to place all the balls into the boxes.

Note it is possible for none of the balls to choose some box, so it is possible that some boxes are empty (but it can't happen that all boxes are empty).