Why is MRS equal to Price ratio?
The mathematical derivation is straightforward: set up a Lagrangian for the utility maximization the consumer solves subject to a monetary constraint, then divide its partial derivatives with respects to the quantities one with another, and there you have it: MRS equals the price ratio. 
The intuition behind it is that, given fixed prices (that is, a price taking situation), the ratio of the marginal utilities must also remain constant: suppose by contradiction this isn't the case, and that some good provides relatively high marginal utility. Then the consumer would profit from deviating to consuming a basket in which the quantity of said good is higher, since the increase in utility would be higher than the costs incurred.