It seems they mention the arboretum many times. Apparently Miles O'Brien proposed to Keiko there. Barclay and Troi go for a walk there. A tree is beamed to the Crystalline Entity from there. But if I recall, it is always off screen.
Do we ever see it? If not, why not? How big is it?
TL;DR: Yes, the Arboretum of the NCC-1701-D was shown in several episodes.
The Arboretum on the Enterprise-D was located on Deck 17, Section 21-Alpha and was roughly three times the size of a standard holodeck. The arboretum was broken up into three sections - a hydroponics/botany lab, a greenhouse, and the main arboretum.
The lab and greenhouse are where Keiko O'Brien spends most of her time, and she can be seen working in these areas several times. For example, she is working in the greenhouse section when Data approaches her during TNG S04E11 "Data's Day" at the request of Miles O'Brien. The lab and greenhouse are located at opposite ends of the room, with the main Arboretum in the middle.
The main section of the Arboretum, which includes numerous plants and a small pond, is what most crew think of as "The Arboretum". It appears in numerous TNG episodes, most notably S05E22 "Imaginary Friend" and S07E07 "Dark Page".
Note: The fourth season episode "Data's Day" was the first appearance of both Keiko AND the Arboretum, as well as several other new sections of the ship - including the barbershop and the replicator center.
There are several episodes of ST:TNG with scenes set in the arboretum. The one that comes to mind specifically is the episode Dark Page (season 7 episode 7) which is summarised here.
It appears on-screen several times, first in Data's Day and then later in both in Imaginary Friend and Dark Page.
Children playing in Arboretum in Imaginary Friend
The Troi women in Arboretum from Dark Page - (this appears to be the reverse angle of the previous shot)
