Others have provided intuitive explanations; I thought I’d provide a short mathematical one.

Suppose a firm faces a production function $q = f(K,L)$. Let the cost of capital (per unit) be $r$ and the cost of labor (per unit) be $w$. 

The firm’s problem is:

$$\max_{K,L} π = pf(K,L) - rK - wL$$

Now, a profit-maximizing firm’s first order conditions are:

$$π_K = pf_K - r = 0 \\ π_L = pf_L - w = 0$$

We can rearrange terms to represent these conditions as:

$$p = \frac{r}{f_K} \\ p = \frac{w}{f_L}$$

Equating the two:

$$\frac{r}{f_K} = \frac{w}{f_L} \implies \frac{f_L}{f_K} = \frac{w}{r} $$

Since $MP_L \equiv f_L$ and $MP_K \equiv f_K$, we get:

$$\frac{MP_L}{MP_K} = \frac{w}{r}$$

Hope that helps.