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
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.