Like shares of public companies traded on the NYSE, the price of a cryptocurrency changes in real-time as supply and demand for a particular cryptocurrency balances (and rebalances). When the amount of demand for a cryptocurrency exceeds the level of supply available, prices rise. When supply exceeds demand, prices fall.
This has to do with mechanics of how orders to buy and sell cryptocurrencies are filled. Imagine you want to buy $100 worth of Bitcoin. When you order Bitcoin, a crypto exchange, like Coinbase, will seek to fill your order by matching it with a sell order for Bitcoin at the lowest price. If there is a high amount of demand, then there may not be enough sellers to satisfy all the orders for Bitcoin. Coinbase would automatically match you with the next lowest price. Through this continuous action, the price ticks up and down 24/7.
Taking a step back, it is important to recognize that macroeconomic trends, media coverage, promotion of particular projects by public figures, and the activity of other investors can drive demand and supply for cryptocurrencies up and down.