There are multiple concepts involved here. Here is a simplified explanation of how it works, assuming you are asking how it happens rather than why which is much harder to answer :-).
Trading a cryptocoin (and a stock, and lots of other things) takes place by using an order book. This is a big list of orders. If you want to buy BTC and are willing to pay $30000 using USD you enter a buy order for $30000 into an exchange's BTC-USD order book. The exchange then looks at the orders from people willing to sell BTC for USD and if anyone will accept $30000 then they take their BTC and give it to you and take your USD and give it to them. This is called "matching" and the orders are said to be "filled" and the act of filling an order is a "trade" (you trade USD for BTC and the seller vice versa). Whether the order is a buy or sell is called the "side", sometimes they are known as bid (buy) and ask (sell) because it is a bit like an auction.
As you can imagine, everyone wants to get the best price. So matching will only happen at the point where buy and sell order prices meet. This means the BTC-USD last trade will always be the most recent price that someone was willing to buy or sell BTC for in USD. So this determines the price of the asset on the exchange. Apps like CoinMarketCap then look at the last trade prices from many exchanges and average them to get the price of the asset they show to you. Prices on different exchanges will almost always converge because if they are different people will make money by buying on the lower and selling on the higher (arbitrage) which will move the prices together.
So the mechanism which determines a currency pair price like BTC-USD is demand - if lots of people want to buy then they will enter higher bid prices (or just take whatever sellers are asking - called a market order), which will make the price rise. If lots of people want to sell they will accept lower prices which will push the price down.
The reason DOGE won't reach $100 is to do with market capitalization. This is a measure of the total money invested in an asset, that is how much every DOGE in the world is worth. That is, it is the number of DOGE multiplied by the price for each. At the moment there are 130 billion DOGE together worth $24 billion at a price of $0.18. If DOGE was $100, the market cap would be $13000 billion (13 trillion) which is more than 800x that of BTC and about 60% of US GDP. You can probably see that reaching this will be realistically impossible.