For someone who wants a deeper understanding of what exactly are we doing with MACD you first need to understand what we are doing using technical indicators in general, & moving averages in particular.

First -

All technical indicators are derivatives -

In the sense that their values are derived from either price or volume of underlying asset. So Moving Averages, RSI, MACD, Stochastics, Average True Range are all indicators. Contrast these with other technical analysis terms you may have heard like Head & Shoulders, Trendlines, Triangles, etc. These are also actively used by technical analysts, but these together are part of a group called technical patterns. These are visual in nature, as compared to indicators which are numerical in nature.

Second -

What are we trying to achieve with indicators?

We are trying to analyse price/volume behavior of a stock in terms of numbers and trying to generate some meaning or a tradeable idea out of them. Converting price action into numbers & then chart the numbers also allows one to get both a numerical & visual feel for price movement.

Third - What do we achieve with moving averages ?

We are trying to filter out noise to develop a better or a general picture of where prices are moving to. Its just like having a look at an index to gauge where prices of various stocks are moving in general. So, moving averages act as filters to remove noise from movements in each timeframe & to be able to see the big picture instead.

Fourth - What are we doing with MACD

When we calculate MACD, we are then doing multiple filter analysis - that is we are running price through multiple filter (instead of just one - in case of moving average) to get an even more refined picture of trend & movement of prices. So when we take the difference of two moving averages & calcute its moving average, we are essentially applying a double filter to prices to smoothen out noise. In mathematical terms, if Moving average is the first derivative, then MACD is the second derivative.