If we are dealing with a two body system and point particles, the the eccentricity is a constant of the orbit.

In practice, there are other bodies in the Solar System and bodies are of finite size. Both interactions with other bodies and tidal forces increase towards perihelion. On average, this extracts energy from the orbit nearer perihelion, which means the object does not reach as large an aphelion and thus reduces the eccentricity.

This is not the reason that planetary orbits are close to circular.