Telegram has all of your chats in readable form on their servers. Any app can connect to it, granted they get a secret token by sending an SMS confirmation code (and optionally a password). Telegram itself has various apps.
When they connect, these apps download your settings, your messages, maybe some contacts, and probably the last few images you’ve sent/received.
You’re not bypassing anything as a third party chat client, you’re using the same information to log in as the official one.
This is how 99% of apps works. Telegram has opted to make the process for connecting open. Other apps try to hide their process, and it can take days to weeks for someone with mild interest to develop an alternative client, though basic send/receive functionality can be built in an hour or so. It’s really the chat app around the protocol that takes a lot of work.
This isn’t always permitted by the terms of service. Discord, for example, tells you they will ban you if you use unofficial apps (they rarely do but it can happen). That’s why every alternative Discord client pretends to be the official one.
Telegram’s openness is one of the reasons I preferred it to WhatsApp, before WhatsApp got itself some basic chat encryption. Telegram never bothered to implement that, so it goes into the “good enough for things I wouldn’t mind leaking too much” pile for me. It’s a real shame, because there are some real fun Telegram bots out there, and being able to automatically send messages through Telegram is a nice touch.
Compare Telegram to websites: web servers host the same type of pages, but any browser on any device can fetch and display those pages, all because the web standard is open. That’s why the web became a success. There were closed-off versions of the web, but they demanded you used their tools and paid for their server software and nobody uses them except for very weird industrial applications.