The short answer is they believe it will go up in price.

But why would a cryptocurrency go up in price?

For the same reason anything goes up in price. Supply and demand.

Why would a cryptocurrency be in demand?

It might help to think about it in the context of another currency, the US dollar.

People say cryptocurrencies aren't backed by anything but the same is true for the USD.

The real thing backing the US dollar is demand.

The USD is in demand because it's useful.

Useful for paying US taxes, but also because it performs the functions of a currency well.

The main functions of a currency are to store value and act as a medium of exchange.

It stores value well because it maintains purchasing power, and it's a good medium of exchange because it's accepted everywhere.

Each function drives demand and reinforces utility in other areas. More demand for USD as a medium of exchange reinforces its ability to store value, and vice versa.

It's a classic network effect.

The more people use it, the more useful it becomes. The more useful it becomes, the more people use it.

The more useful it is, the more demand it has, the higher it's value.

Money is similar to language in terms of network effects.

Everyone has their local version but use an international one as a common frame of reference. The USD is the reserve currency of the world like English is the reserve language of the world.

Bitcoin proponents think that Bitcoin will become the world's reserve currency. If it's in high demand and has a supposed limited supply, the value should go way up.

The question becomes: if value is driven by demand, and demand is driven by usefulness, are cryptocurrencies useful?