There are a few reasons I've heard/read before for teacups being so small.

1: Teacups are small so you enjoy the flavour of the tea. A good quality tea should infuse with water quickly, so you want to quickly put your leaves in, take them back out, and then have your nice little taste of tea at the ready. This method, known as Gong-fu, usually used cups about 30ml and multiple infusions to get every taste out of your tea. Some flavours don't come out until later than others, meaning if you did one long infusion in a big cup, you might end up with one big mix of flavours that don't stand out. By doing repeated smaller infusion, you can often taste different flavours in every cup, even though you're using the same leaves/blend. For example, using one blend your first infusion may taste like a regular green tea, but your second may have hints of lemongrass or ginseng, which then become the main flavour in your third infusion.

2: Having Tea was a highly social event in much of the far East. By keeping cups small you ensured people weren't constantly guzzling down huge cups of tea mid conversation, so everything flowed much better. Shopkeepers would often serve tea to customers, so while the Shopkeeper made his pitch, the customer could have a drink, then while the customer replied, the Shopkeeper could have a drink. Keeping the cups small added to the natural rhythm of conversation, with a small cup helping keep you going during long conversations and without it going cold in the gaps between. A small cup also takes a lot less time to prepare meaning you could quickly have a few cups during a conversation, with the host using lulls in the conversation to refill everyone, helping avoid awkward silences. Small cups also mean you can fill more cups from a single teapot, allowing a host to potentially serve more people than if they were doing so with larger cups. It's more polite to serve everyone a small cup than it is to serve half your guests a big cup, then have to leave to brew more tea for the other half of them.

3: British/European Tea Cups were originally small as a copy of Oriental Tea Cups, but size became a part of British Tea making/drinking tradition too. Though larger than their Eastern counterparts, British Tea Cups used to be small because when you served tea it would be out of a pot. You'd pour everyone out a small cup of tea and they'd drink that, then you'd give them a refill. Teapots were usually better insulated than the cups, so less heat would be lost from your tea by leaving more in the pot. If you poured it all out into larger cups or mugs, there's the chance that your drink would go cold. You're better having multiple refills in a smaller cup than one big one that goes cold halfway through drinking it. Small cups also cool faster so you can have the drink you pour sooner, rather than waiting a long time for your drink to cool enough to even have a sip.

4: Porcelain was expensive. Getting good quality cups could be difficult, and if you didn't want to spend a stupid amount of money on large cups of the same quality, it was better to get smaller cups of an assuredly good quality than it was to buy larger cups that may be poorly made to keep them at a lower price. This is probably the least important reason of all nowadays, but thousands or even hundreds of years ago in places like rural China this was definitely a larger factor than it is now.

I hope this helps answer your question. It's largely a matter of etiquette and practicality, and it can be hard to find sources for these. Many of the things I've read related aren't available online and were instead actually in tearooms and archives I've visited, as little handouts or pamphlets.

Sources:

TeaAngle: What's with the small tea cups?

The Classic of Tea - a Chinese text from around 760CE. The first known monographic text that documents tea preparation, including the Gong-Fu method.

English Heritage: How to do Afternoon Tea like the Victorians

Edit: some grammar and formatting.