"The day is beautiful."
Translation:La tago estas belega.
According to the Wikipedia article called Esperanto/Appendix/Table of affixes, -eg- is the augmentative suffix. Basically, the -eg- just makes things, "bigger," if you know what I mean.
The examples from the article are:
domego (a mansion, from 'house');
virego (a giant, from 'man');
librego (a tome, from 'book');
varmega (boiling hot);
ridegi (to guffaw, from 'laugh');
ega (great, humongous)
The opposite of -eg- is -ef-, the diminutive affix, which makes things, "smaller."
From the same article, the examples are:
dometo (a hut, from 'house');
libreto (a booklet);
rideti (to smile, from 'laugh');
rompeti (to crack, fracture, from 'break');
boleti (to simmer, from 'boil');
eta (small, little)
No, "to make beautiful" is "beligi". "To become beautiful" would be "beliĝi".
"Beli" is simply "to be beautiful". It's the case for all verbs derived from adjectives.
Another example is "blanka" ("white"). The verb "blanki" just means "to be white".
"To make (something) white" would be "blankigi", and "to become white" would be "blankiĝi". Note that in English, both are translated by "to whiten".