How does the GDP multiplier work? When we learn about the GDP multiplier, it is portrayed as 1/MPS, because savings are supposedly a leak out of consumer spending. However, what I don't understand is aren't those savings spent anyways in the product market? Don't they go to the financial market and get used by businesses to invest in projects? At that point the GDP multiplier would be infinite and the only thing that would matter is how fast people spend. Am I wrong in thinking this?
The multiplier comes from the solution to the goods market equilibrium. In economics everything is endogenous. Increase in income increases consumption that increases demand, demand increases production and production increases income. However, as an echo in a cave the initial increase in income gets 'weaker' as it cycles through the economy and the result 1/MPS tells you how much after accounting for the above mentioned cycle 1e increase in for example government spending increases the output. 

Actually on fundamental level its not about marginal propensity to save but marginal propensity to consume (MPC), (however note that MPS=1-MPC) which is the channel through which increase in income affects the consumption which then affects demand and output.

A standard undergraduate textbook level consumption function looks something like this:

$$C = c_0 + c_1 (Y-T)$$

Where $C$ is the consumption, $c_0$ autonomous spending $c_1$ is the MPC and $Y-T$ is the disposable income ($Y$ is the income/output and $T$ taxes).  Mathematically, $c_1$ or MPC is what determines how $C$ changes when income changes. However, in order to see how much 1e increases output we cannot look at just MPC because as mentioned in the first paragraph the 1e cycles in the economy. So you actually need to calculate the geometric sum of the consumption multiplier being applied over and over again. Mathematically:

$$1+c_1+c_1^2 + ... c_1^n = \frac{1}{1-c_1}$$

Even though the 1e can be thought of circulating in economy infinitely  many times that does not mean it increases the output by infinity as many infinite sums actually converge to some actual value. In this case as long as $c_1$ is less than one, which it should as its the proportion of consumed income, the infinite sum would converge to some finite value not diverge to infinity. 

However, while the above explains the logic behind why the multiplier is what it is it probably does not answer your question about saving. To see what role savings plays in the model consider the standard goods market equilibrium which is found by first substituting $C$ into the GDP identity and then solving for $Y$ (which is where you actually get the multiplier from):

$$Y = \frac{1}{1-c_1} \left(c_0  +I+G-c_1T\right)$$

Where as mentioned above $Y$ is output $1/(1-c_1)$ is 1/MPS $c_0$ autonomous spending, $I$ investment $G$ government spending $c_1 T$ are taxes multiplied by MPC. Now saving actually hides in the model in the investment $I$. By definition investment must be the sum of private and public saving, assuming away public saving for the sake of simplicity we can say that investment is equal to private saving so $I=S$. Consequently the multiplier 1/MPS affects also saving and when you increase the amount of savings the same multiplier is applied as to government spending or autonomous consumption so you are completely correct in wondering about that.

However, the reason why 1/MPS 'the multiplier' is that, as already mentioned, in this model the channel through which multiplier works is the effect of income on consumption which then affects all the rest.

Also note that in a short run $I$ can be considered fixed $\bar{I}$, I wont go into full detail about why that is as it is outside the scope of your question but basically in short run during recession investment is not always responsive to increase in savings because any increase in savings will be offset by changes to other variables. A consequence of assuming that investment is fixed and does not change with the level of savings then the savings will have no multiplier and you will get the "paradox of thrift".