Why is Curve25519 mostly used for key exchange? When i studied the Applications where Curve25519 is used, i found out, that it is mostly used for the key exchange. Examples are the Signal Protocol and Threema.
I know, that Curve25519 has a pretty fast arithmetic, so my first idea was, that it is used to decrease the server load. Am i right with my idea? Can someone give me a better answer?

Edit:
I understand that my question may be not clear enough: Why is it not used for encryption/decryption of messages, when used in messenger like Threema?
A number of reasons contribute to this.

Curve25519 has a non-governmental origin. 

It's a curve that's very safe by design, and impregnable to many side-channel and other weaknesses that other curves suffer from.

Also, it's a curve with 'nothing up my sleeve' coefficients.

Unlike the NSA curves, which NIST endorse.

Although not directly related, after the backdoor in Dual_EC_DRBG had been exposed, suspicious aspects of the NIST's P curve constants led to concerns that the NSA had chosen values that gave them an advantage in finding private keys. Since then, many protocols and programs started to use Curve25519 as an alternative to NIST P-256 curve.

As Kelalaka states, the constant time scalar multiplication provided by the Montgomery ladder is also pretty cool.

You cannot directly encrypt arbitrary amounts of data direct with elliptic curves. 

The public key is a point on the curve and the private key is a number.

There isn't an elegant mechanism to directly encrypt (generalised ElGamal notwithstanding).

That's why key agreement revolves around finding a shared point on the curve and using a digest of the x co-ordinate of that shared point to use with a fast, secure symmetric cipher.