You will find many different reasons for people to collect rare coins. The reasons are as varied as you might imagine, and each individual collector will likely have slightly different reasons to collect.
Some people enjoy the history associated with the coins. You will be able to find coins that are hundreds of years old, and in some cases much older. Those who enjoy history will be able to appreciate the historic and cultural value of the coins as well as their monetary value.
Rare coins will often have a high value as well. Some choose to collect the coins for their monetary value and they look at them from the perspective of an investor.
Others will collect coins because of their beauty. Many of the coins that you can find have interesting and intricate designs on them. The artwork is quite impressive in many cases, and most people will be able to appreciate the craftsmanship that went into them.
Some people simply like to collect things, and coins make the perfect hobby. You can find coins that suit your particular likes. Completing a set of coins from a specific era or from a specific mint is a fun adventure.
Most people will find a balance of each of the above reasons when they start collecting coins. Everyone will find something to love about the hobby of collecting rare coins, and that is why it has grown in popularity in recent years.