Why does savings equal investment (scenario)? Scenario 1: There's an economy of two people (Joe and Amanda). Joe buys a 500 dollar car from Amanda (which she made herself from raw materials in her back yard). Amanda takes the 500 dollars and hides it under her bed. Savings = 500 dollars, Investment = 0 dollars. What am I missing?

Scenario 2: Same Amanda and Joe. Joe gets 500 dollars from the Federal Reserve (they just gave it to him). He puts it in a bank. The bank decides to only lend out 250 dollars of it. Amanda invests in inventory. S = 500, I = 250. Right?

These are two cases where investments do not seem to equal savings. How can these cases be explained?
To clarify the first scenario: you assume Joe can buy this car. However, Joe has to either earn an income (and therefore, produce something himself) or he has to dissave. In the first case, Joe has produced 500 output and earned an income of 500. He spends his income on a car, but Amanda saves the money. As is clarified in other answers, Joe's output goes to inventories (because not purchased by Amanda), which are counted as (unplanned) investments. S = I = 500, C = 500, Y = S + C = 1000. You mention in a comment that most consumers do not produce something. In fact, they do. At work, when earning their income (with which they buy the car), they created added value (this is the production (500) that goes to the inventory when Amanda does not buy it). Think of the income they earn at work as their part in the created added value, their contribution to real production. Consider the second case when Joe does not work.

In the case Joe dissaves to buy the car instead of working and using his income (he just drives around, without producing), 500 savings have to be deducted from your final result for the first scenario, so that S = I = 0, C = 500, Y = S + C = 500.

In the second scenario, forget about Joe's 500 for a while. Assume Amanda invests 250 (one car) in inventory, which means she has produced 250 (one car) first (and sold nothing), so that I = Y - C = 250, C = 0. Consumption is zero because Amanda sold nothing and Joe bought nothing. 

An income can only be earned when value added was produced. Therefore, the  500 Joe has received cannot be seen as an income, and since only income can lead to consumption or saving, Joe had neither consumed nor saved. The 500 Joe received is an expansion of money supply (and has no real interpretation) and will increase prices proportionally (assume V and Y to be constant in the fisher equation (MV=PY)). The 250 that Amanda produced has now increased in value, but in real terms her investment in inventory will still refer to the 250 (one car). 

This actually illustrates the importance of distinguishing nominal and real variables in the economy. The macro-economic identity you refer to is only about real variables.