Why is key management a major drawback of symmetric cryptography in a network of users?

Each pair of communicating parties needs its own shared key — so for n users the total number of keys grows as n(n-1)/2, quadratically.

Example sizes:

And every key must be:

• Generated securely (CSPRNG).
• Exchanged over a secure channel (chicken-and-egg).
• Stored securely on every endpoint.
• Rotated periodically.
• Revoked if a participant leaves or is compromised.

The bigger question this raises:

"Und wie wird der Schlüssel sicher ausgetauscht?" (And how is the key securely exchanged?)

That's literally the question asymmetric cryptography was invented to answer. Two solutions:

1. Key Distribution Centre (KDC) — a trusted central server hands out session keys. Kerberos works this way. Still needs each user to pre-share a key with the KDC.
2. Public-key / Diffie-Hellman key exchange — derive a shared secret over a public channel without prior contact. The breakthrough of 1976 that made modern HTTPS possible.

Tip: Pure symmetric setups work fine for small fixed groups (e.g. point-to-point VPN, satellite uplink). For the internet's billions of arbitrary peers, you need asymmetric crypto — and that's exactly what TLS does to bootstrap a session.